How does ice melt when immersed in water? When an ice cube is immersed in water at a room temperature, how is the thermal energy from the water transferred to the ice cube?
Currently I have two answers:


*

*Infrared radiation from the water transfers thermal energy to the ice cube, which increases the ice cube particles KE store, breaking the intermolecular bonds of the ice cube, melting it.

*The Brownian motion of the water particles  causes them to collide with the ice cube, transferring KE to the ice cubes particles, increasing temperature, breaking intermolecular bonds and melting it.
 A: Energy transfer methods
In general, there exist three heat transfer mechanisms:


*

*Thermal radiation transfers heat across a distance. More accurately, it is the transfer of wavelengths on the spectrum of light that when absorbed by the body is converted into heat). It follows Stefan-Boltzmann's law: $$\dot q_\text{rad}=\varepsilon\sigma_sA(T_1^4-T_2^4)$$ ($\dot q$ is energy per second transferred from body 1 to body 2, $T$ temperature, $\varepsilon$ emissivity, $\sigma$ the Stefan-Boltzmann constant, $A$ the radiating surface area.)

*Thermal conduction transfers heat through a solid. It is defined for a continuum, a solid material, but can be thought of as heat passed on between neighbour particles. It follows Fourier's law: $$\dot q_\text{cond}=A\kappa\frac{\Delta T}{\Delta x}$$ ($A$ is area through which the heat flows, $\kappa$ thermal conductivity, $\Delta T$ temperature difference between two points, $\Delta x$ distance between those two points over which the heat is tranferred.)
When you mention Brownian motion, it is relevant here with conduction: The random motion of particles, electrons etc. cause them to "bump into" and interact with neighbour particles. If one particles is more energetic, at a collision between particles they will share some of the kinetic energy. This is how thermal energy is conductively transferred.


*

*Thermal convection transfers heat to/from a body by flowing close to it and deliver/absorb thermal energy to/from the surface. In some sense, it can be thought of as conduction between a fluid particle and a surface particle, where the fluid particle right after is replaced with a new, fresh one. Delivery/absorption of thermal energy from a single fluid particle is negligible as it carries a very little amount of energy, but with constant replacement of particles with newer ones, the energy transferred accumulates and becomes significant. This fluid-in-motion-induced heating/cooling effect is termed convection. It follows the relationship: $$\dot q_\text{conv}=Ah(T_\text{fluid}-T_\text{body})$$ $A$ is area exposed to the fluid. $h$ is the heat transfer coefficient and it highly depends on the scenario (the fluid, the flow, the surface interaction etc). $h$ is often experimentally determined beforehand.


There are two types of thermal convection: 


*

*Natural convection caused purely by natural factors such as differences in temperature or density (the cooling water near the ice surface becomes denser and sinks, and is thus replaced by other warmer fluid molecules. In general, natural convection is the mechanism behind hot air rising and cold air falling and similar phenomena.)

*Forced convection, which is fluid flow caused by non-natural mechanisms such as by a pump.
In your case we have natural convection: The water particles near the ice surface deliver heat to the ice and in turn cool down. These now "colder" water particles are denser or "heavier" and will sink. New, warmer particles will take their place, ready to deliver more energy to the ice surface and repeat the process.
Which is more dominant?
The above three energy transfer factors are all the possibilities there are to transport energy. They are generally considered on equal terms as three distinct mechanisms with each their own energy transfer models. But, as you can see, convection is basically a "flow-version" of conduction if we consider it microscopically.


*

*For thin fluids (with low viscosity), the convective effect of effective heating/cooling due to fluid motion is dominant.

*For very thick fluids (with very high viscosity), so thick that you might mistake them for solids, heat can flow from particle to particle in a conductive manner, and conduction is dominant.

*For some-what thick fluids, we may see a mix of these factors. The higher the heat capacity (corresponding to lower $\kappa$) of the fluid, the weaker is the conductive mechanism.


In your case with water that has a rather low $\kappa$, we should be able to assume only a predominantly convective mechanism and no/negligible conduction over longer distances in the water. Thermal radiation could still be a factor as well, but at fairly low temperatures, radiation is low (note the power of 4 in the model) and possibly negligible. We end up with only convection (natural in your case) having a large influence in your case - in fluids, this is often the only effect that is relevant to consider, unless when sinking a glowing-hot metal into a very volatile liquid. 
This analysis can be verified by looking up numbers, as some comments ask for, of water and ice for the different models as well by comparing with the viscosity. I will not do this in this answer, but it should be fairly easy to find online; other answers are giving some of such numbers to justify the conclusion.
A: Thermal energy transfer is in the form of heat from the water to the ice cube by natural convection.
If the cube and water together form an isolated system (no heat transfer between them and their surroundings) the heat transfer will continue until all the ice is melted, or until the water temperature equals 0 C at which point any ice remaining will be in two phase thermal equilibrium with the water.
Hope this helps
A: I am in complete disagreement with previous answers which consider convection as the main mechanism for heat transfer from liquid water to the ice cube.
Convection is an important and dominant mechanism to maintain the liquid layers close to the ice surface at higher temperature. Thus, its main role is to ensure that at the surface between liquid and solid a constant difference of temperature is maintained. However, as a mechanism to carry energy from the liquid into the solid, convection simply does not exist! Unless one would think of fluid streams penetrating into the solid, which is not the case.
Therefore we are left with conduction or radiation as possible ways to tranfer thermal energy from liquid water to the ice. A simple order-of-magnitude estimate, based on the formulae of the Stefan-Boltzmann's law and Fourier's law, taking into account the SI values of about $10^{-7}$ for $\sigma_s$, of about $2$ for $\kappa$ of ice, the values of the two temperatures and a value of $\Delta  x$ of the order of a few interatomic distances, shows that the radiation contribution is negligible.
An additional remark could be added on the microscopic description of the melting process.
It is a well established observation that pre-melting, i.e. the melting of a solid starting from the surface layers, instead of than from the bulk, is a phenomenon present even in the case of ice. This observation would exclude the possibility that the melting process in the present case could start in the bulk of the ice.
A: Heat Transfer Modes
The three forms of heat transfer between a system and the surroundings are as follows:
Conduction
This is the transport of heat by particles exchanging their internal energy. It occurs by one of three modes -- molecular collisions (gases), collisions/vibrations (local in liquids and lattice in solids), and free electron transport (in conductors and semiconductors). Conduction requires (or sets up) a temperature gradient in the material that is transporting the heat.
Convection
This is the transport of heat content by the bulk motion of a fluid over an object. It occurs in one of two modes -- free or forced. In free convection, the fluid moves because it is subject to a buoyancy force. In forced convection, we push the fluid. Convection requires a temperature difference. Convection can be modeled using principles of conduction across a film between the fluid and the object.
Radiation
This is the transport of energy from an object as electromagnetic radiation. Radiation only requires that objects have a temperature.
The Melting Process
To melt, atoms in a solid must gain enough energy to leave their bonds in the solid. Fusion is endothermic.
The energy arrives as heat from the surroundings. It arrives by the motion of the hotter liquid water molecules hitting the colder solid. The energy difference between moving liquid molecules and static (vibrating) solid molecules is a temperature difference in internal energy coordinates. That temperature difference needs only be infinitesimal to support the flow of heat from hot to cold. Liquid water does not support free electrons (of course not!) nor does it support lattice vibrations (that is what is happening in the ice). So, the one mode of transport of heat is conduction by molecular collisions from liquid water to solid ice.
The energy as heat can arrive by convection flow. When the system is in a gravitational field, and when the liquid immediately around the ice might become colder than the bulk water, the colder water will be denser. It will start to flow downward by natural convection. Thus, natural convection can be a factor in the heat flow. When the ice is floating on the water (typical), the colder water below the ice will fall down in the warmer water below it. As an inverse case, when you could put the ice cube at the bottom of a container and have hot water above it, you will shut down the natural convection mode. Think also about a cold penny that sits inserted into an insulated floor with hot air above it. The penny will have no natural convection modes because the cold air that might form around it is already denser than the hot air above it. This same thought is behind the formation of cold and hot fronts with thunderstorms in weather patterns.
You did not say whether the tank was stirred. So we can ignore forced convection.
The ice is radiating from it. The hotter water is radiating to the ice. The net radiation flux is to the ice from the water.
Estimates of Magnitudes
The temperatures of the solid ice and liquid water control the net radiation flux. When the liquid is only infinitesimally above the ice in temperature, the net radiation flux is ... small. Add to this that both ice and water have emissivities well below unity and their emissivities are comparable. At the end, you can pretty much say radiation is ... to be neglected.
Natural convection, when it occurs, swamps conduction heat transfer (well, not literally of course). Presuming the ice is at the top allows for this. Saying the ice is surrounding by water and mixed with it will lower its contribution.
At the end, we have conduction. Those "hotter" liquid water molecules are colliding constantly with those "colder" solid ice molecules (hot and cold as measures of internal energy). The transfer of heat is occurring constantly. A reference graph showing the variations in conductivity is found at this link.
Remaining Clarification
In pure materials (water), fusion occurs at a constant temperature. Never, ever can you discuss fusion as a process where the solid becomes hotter. The solid ice in this case stays pinned at one temperature as it completely melts. Inversely, you might find that when you mistakenly think the ice gets hotter during melting, you will immediately have to shut down any and all net heat transfer from the surroundings (liquid) to the system (ice). It is the second law of thermodynamics at play.
