Umbilical Constant Mean Curvature (CMC) slices, i.e., slices where the extrinsic curvature is pure constant trace, K, through the Schwarzschild solution run from future null infinity to future null infinity. The intrinsic metric can be written as $dS^2 = {dR^2 /over 1 - 2M/R + K^2R^2/9} + R^2 d/Omega^2$ where $R$ is the Schwarzschild (or areal) radius, and $M$ is the Schwarzschild mass. Each such slice has constant negative scalar curvature because the Hamiltonian constraint reduces to $^{(3)}R + 2K^2/3 = 0$. All spherical metrics are conformally flat and the equation for the conformal factor becomes $/nabla^2 /phi -C/phi^5 = 0$. I will construct explicitly ALL solutions to this equation and show their relationship to the umbilical slices through Schwarzschild. Unfortunately, the solutions, except in a special case, have to be written in terms of Weierstrass elliptic functions.