  A smooth end of a Bryant surface is a conformally immersed punctured disc of
mean curvature 1 in hyperbolic space that extends smoothly through the ideal
boundary. The Bryant representation of a smooth end is well defined on the
punctured disc and has a pole at the puncture. The Willmore energy of compact
Bryant surfaces with smooth ends is quantized. It equals 4Pi times the total
pole order of its Bryant representation. The possible Willmore energies of
Bryant spheres with smooth ends are 4Pi*d where d is a positive integer
different from 2,3,5,7. Bryant spheres with smooth ends are examples of soliton
spheres, a class of rational conformal immersions of the sphere which also
includes Willmore spheres in the conformal 3-sphere. We give explicit examples
of Bryant spheres with an arbitrary number of smooth ends. We conclude the
paper by showing that Bryant's quartic differential Q vanishes identically for
a compact surface in the 3-sphere if and only if it is the compactification of
either a complete finite total curvature Euclidean minimal surface with planar
ends or a compact Bryant surface with smooth ends.
