What happens when a body of water and a pice of ice are placed in a pricessly controlled zero degree environment
My understanding is, if you put ice into a >0 degree environment it
  will be 0 degrees as it melts and if you put water into a &lt;0 degree
  environment it will 0 degrees as it freezes.


Correct. If you place ice in a container where the temperature is above  $0\ \mathrm{^\circ C}$, the ice will first heat up to $0\ \mathrm{^\circ C}$, and then undergo phase transition to a liquid. The same works the other way around, as you stated. Placing water in a sub-zero temperature will cause the water to cool down to $0\ \mathrm{^\circ C}$ and then it will freeze (form a solid). See picture below:




  But want happens if you put water and ice in an environment maintained exactly at 0 degrees.


This really depends on the other qualities of water. If you add salt in to the water, it will remain as a liquid at $0\ \mathrm{^\circ C}$. This is due to freezing-point depression, which is described reasonably well in Wikipedia.

If you are considering pure water (no salt or other solvents mixed), then you will get an equilibrium of water and ice. It might be easier to understand, if you keep in mind that although the overall temperature is $0\ \mathrm{^\circ C}$, there will be some variation between single molecules, some having a lower amount of thermal energy and some having a higher amount.