  The phase diagram of the unconventional superconductor UPt$_3$ is explained
under the long-standing hypothesis that the pair wavefunction belongs to the
$E_{1g}$ representation of the point group. The main objection to this theory
has been that it disagrees with the experimental phase diagram when a field is
applied along the c-axis. By a careful analysis of the free energy this
objection is shown to be incorrect. This singlet theory also explains the
unusual anisotropy in the upper critical field curves, often thought to
indicate a triplet pair function.
