  The present PREX-II and CREX experiments are measuring the rms radius of the
weak charge density of $^{208}$Pb and $^{48}$Ca. We discuss the feasibility of
a new parity violating electron scattering experiment to measure the surface
thickness of the weak charge density of a heavy nucleus. Once PREX-II and CREX
have constrained weak radii, an additional parity violating measurement at a
momentum transfer near 0.76 fm$^{-1}$ for $^{208}$Pb or 1.28 fm$^{-1}$ for
$^{48}$Ca can determine the surface thickness.
