  We study the thermal conductivity within the E$_{1g}$ and E$_{2u}$ models for
superconductivity in UPt$_3$ and compare the theoretical results for electronic
heat transport with recently measured results reported by Lussier, Ellman and
Taillefer. The existing data down to $T/T_c\approx 0.1$ provides convincing
evidence for the presence of both line and point nodes in the gap, but the data
can be accounted for either by an E$_{1g}$ or E$_{2u}$ order parameter. We
discuss the features of the pairing symmetry, Fermi surface, and excitation
spectrum that are reflected in the thermal conductivity at very low
temperatures. Significant differences between the E$_{1g}$ and E$_{2u}$ models
are predicted to develop at excitation energies below the bandwidth of the
impurity-induced Andreev bound states. The zero-temperature limit of the $\bf
\hat c$ axis thermal conductivity, $\lim_{T\to 0} \kappa_c/T$, is
\underline{universal} for the E$_{2u}$ model, but non-universal for the
E$_{1g}$ model. Thus, impurity concentration studies at very low temperatures
should differentiate between the nodal structures of the E$_{2u}$ and E$_{1g}$
models.
