What happens when ice and water are placed together at the same temperature in an isolated system?

Suppose ice and water (not necessarily of the same mass) are both at $0^{circ}$C and placed into a perfectly insulated, sealed container so that it is filled completely. If we leave this system alone for some time, what happens?

I think this may be similar to what happens when ice melts or water freezes, as then there are different states of matter at the same temperature. Thus, my best guess is that both the amount of water and the amount of ice will remain constant, as there is no heat flowing into or out of the system to drive the phase change either way.