Heat Transfer Modes
The three forms of heat transfer between a system and the surroundings are as follows:
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.
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.
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.
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.