  To establish the mechanism of unconventional superconductivity in
Sr$_2$RUO$_4$, a prerequisite is direct information concerning the
momentum-space structure of the energy gaps $\Delta_i(k)$, and in particular
whether the pairing strength is stronger ("dominant") on the quasi-1D ($\alpha$
and $\beta$) or on the quasi-2D ($\gamma$) Fermi surfaces. We present scanning
tunneling microscopy (STM) measurements of the density-of-states spectra in the
superconducting state of Sr$_2$RuO$_4$ for $0.1 T_C<T<T_C$, and analyze them,
along with published thermodynamic data, using a simple phenomenological model.
We show that our observation of a single superconducting gap scale with maximum
value $2\Delta \approx 5 T_C$ along with a spectral shape indicative of line
nodes is consistent, within a weak-coupling model, with magnetically mediated
odd-parity superconductivity generated by dominant, near-nodal Cooper pairing
on the $\alpha$ and $\beta$ bands.
