  We obtain a phase diagram for a trapped two-dimensional ultra-cold Bose gas.
We find a critical temperature above which the free energy of a state with a
pair of vortices of opposite circulation is lower than the one for a
vortex-free Bose-Einstein condensed ground state. We identify three distinct
phases which are, in order of increasing temperature; a phase coherent
Bose-Einstein condensate, a vortex pair plasma with a fluctuating condensate
phase, and a thermal Bose gas. The existence of the vortex pair phase could be
verified using current experimental setups.
