Mathematically, a normalized (unit trace) density matrix describes a pure state only if $1,0$ are the only eigenvalues that the density matrix has.
Physically, there can't be any objective answer to the question whether a physical system is in a pure state or a mixed state. The state vector or the density matrix aren't classical degrees of freedom; instead, they are collections of complex numbers that encode subjective probabilistic knowledge about the system.
A system in a pure state – any pure state – represents the maximum knowledge one may have about the system in quantum mechanics. It is an analogy of the point on the phase space in classical physics. A system in the mixed state is always just a probabilistic mixture of pure states,
$$ \rho = \sum_i p_i |i\rangle\langle i|$$
for some orthogonal ket vectors, and this mixed state is a counterpart of probabilistic distributions on phase space in classical physics. We use several ket vectors with $p_i$ strictly in between $0$ and $1$ because we don't know which one is realized. But in principle, we could know. Perhaps, there exists another observer who does know and who describes the situation by a pure state.
For example, we may talk about a photon (with a known momentum) coming from a light bulb. If we know its linear polarization, we will describe it by a pure state. If we don't know the polarization, we may describe it by a mixed state. But it may be the same photon and other people may know less or more about the photon's polarization.
So there can't exist any operational procedure that objectively settles the pure/mixed question. The answer depends on one's knowledge – an important and universal fact about any quantum mechanical theory.
Incidentally, the comments about the temperature are completely unrelated to the original question. It is not sensible to associate a temperature with a single degree of freedom. So if a box is a microscopic state and not a macroscopic piece of wood, it doesn't have any temperature. Nevertheless, multi-body systems with many particles may be found in microcanonical, canonical, or grand canonical ensembles. When we talk about many particles, it is possible to reconstruct the distribution function and it does become measurable – in this sense, the ensembles are distinguishable by a measurement. However, if we talk about a single particle, it again makes no sense to physically ask whether it obeys a microcanonical or canonical distribution. These are two possible density matrices that the particle may pick and different density matrices describe different subjective knowledge about the system. For one observer, a particular electron may be in a microcanonical ensemble, for another observer, it may be in a canonical ensemble.