  The phase diagram of the superfluid phases of $^{3}$He in 98% aerogel was
determined in the range of pressure from 15 to 33 bars and for fields up to 3
kG using high-frequency sound. The superfluid transition in aerogel at 33.4
bars is field independent from 0 to 5 kG and shows no evidence of an
$A_{1}-A_{2}$ splitting. The first-order transition between the A and B-phases
is suppressed by a magnetic field, and exhibits strong supercooling at high
pressures. We show that the equilibrium phase in zero applied field is the
B-phase with at most a region of A-phase {\small $\alt$} 20 $\mu$K just below
T${_c}$ at a pressure of 33.4 bars. This is in contrast to pure $^{3}$He which
has a large stable region of A-phase and a polycritical point. The quadratic
coefficient for magnetic field suppression of the AB-transition,
$g_{a}(\beta)$, was obtained. The pressure dependence of $g_{a}(\beta)$ is
markedly different from that for the pure superfluid, $g_{0}(\beta)$, which
diverges at a polycritical pressure of 21 bars. We compare our results with
calculations from the homogeneous scattering model for $g_{a}(\beta)$, defined
in a Ginzburg-Landau theory in terms of strong-coupling parameters $\beta$. We
find qualitatively good agreement with the experiment if the strong-coupling
corrections are rescaled from known values of the $\beta$'s for pure $^{3}$He,
reduced by the suppression of the superfluid transition temperature. The
calculations indicate that the polycritical pressure in the aerogel system is
displaced well above the melting pressure and out of experimental reach. We
cannot account for the puzzling supercooling of the aerogel AB-transition in
zero applied field within the framework of known nucleation scenarios.
