We discuss dissipative dynamics of a vortex state in a trapped Bose-condensed gas at finite temperatures and draw a scenario of decay of this state in a static trap. The interaction of the vortex with the thermal cloud transfers energy from the vortex to the cloud and induces the motion of the vortex core to the border of the condensate. Once the vortex reaches the border, it immediately decays through the creation of phonons. We calculate the characteristic life-time of a vortex state and address the question of how the dissipative dynamics can be studied experimentally.