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 $2Delta approx 5 T_C$ 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.
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