  The dissipative dynamics of a vortex in a finite temperature trapped
Bose-Einstein condensate are shown to be governed by a {\em diffusive
instability}. In the weakly interacting regime we find a cross-over from
instability to metastability of the vortex, determined only by the relative
rotation rate of the vortex and thermal cloud. Even in the thermodynamically
stable regime there is a finite lifetime arising from diffusive instability. We
find the mean exit time for the vortex from the condensate, and show that near
the critical frequency for thermodynamic stability the lifetime has a universal
form which is independent of the condensate size and the vortex precession
frequency. In the thermodynamically stable regime the steady-state two-time
position correlation predicts `thermal bunching', giving an increased
probability of positional coincidence for short times.
