  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 $\rm
La_{2-x}Sr_xCuO_4$ (LSCO) 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 $H^*(T)\ll H_{c2}(T)$
is shown to be the field scale for the flux-flow resistivity.
