Ice in contact with a heat reservoir at $0^\circ \textrm{C}$?

TL;DR: If the ice has surfaces then it will completely melt to form water. If the ice has no surfaces (it's infinitely large) then it will remain completely ice.

Here's the reasoning:

If we imagine an idealised system that is infinitely large in extent (i.e. no surfaces and therefore no heterogeneous nucleation) then the ice will remain in an ice state, with small water nuclei that spontaneously form and then disappear again. To see why, consider how the free energy changes when a water nucleus of radius $r$ forms:

$$ \Delta G=-\Delta G_V (4\pi r^3/3) + \gamma(4\pi r^2) $$

where $\Delta G_V$ is the difference in free energy between the bulk phases, and $\gamma$ is the surface free energy. At the coexistence point, $\Delta G_V=0$ by definition, and so the the growth of a water phase will only ever cost energy (i.e. there's no critical size). The size distribution of these random water droplets will be given by the Boltzmann distribution:

$$ P\propto \exp(-4\pi r^2 \gamma/k_BT) $$

In the real world, the ice will have surfaces. And the surface energy of ice is greater than that of water, so the outer layers will melt, resulting in a block of ice surrounded by a layer of water. As before, since neither bulk phase is preferred, then the system will move towards the state that minimises the interfacial free energy, and so the ice in the core will shrink in size until it disappears.

The result is pure water with tiny ice nuclei spontaneously forming and melting with the same size distribution as before ($P\propto \exp(-4\pi r^2 \gamma/k_BT)$).