  The phase diagram of superconducting UPt$_3$ is explained in a
Ginzburg-Landau theory starting from the hypothesis that the order parameter is
a pseudo-spin singlet which transforms according to the $E_{1g}$ representation
of the $D_{6h}$ point group. We show how to compute the positions of the phase
boundaries both when the applied field is in the basal plane and when it is
along the c-axis. The experimental phase diagrams as determined by longitudinal
sound velocity data can be fit using a single set of parameters. In particular
the crossing of the upper critical field curves for the two field directions
and the apparent isotropy of the phase diagram are reproduced. The former is a
result of the magnetic properties of UPt$_3$ and their contribution to the free
energy in the superconducting state. The latter is a consequence of an
approximate particle-hole symmetry. Finally we extend the theory to finite
pressure and show that, in contrast to other models, the $E_{1g}$ model
explains the observed pressure dependence of the phase boundaries.
