Measurements of the Nernst signal in the vortex-liquid state of the cuprates to high fields (33 T) reveal that vorticity extends to very high fields even close to the zero-field critical temperature T(c0). In overdoped La2-xSrxCuO4, we show that the upper critical field H(c2)(T) curve does not end at T(c0), but at a much higher temperature. These results imply that T(c0) corresponds to a loss in phase rigidity rather than a vanishing of the pairing amplitude. An intermediate field H*(T)<<H(c2)(T) is shown to be the field scale for the flux-flow resistivity.