  Sr$_2$RuO$_4$ has stood as the leading candidate for a spin-triplet
superconductor for 26 years. Recent NMR experiments have cast doubt on this
candidacy, however, and it is difficult to find a theory of superconductivity
that is consistent with all experiments. What is needed are symmetry-based
experiments that can rule out broad classes of possible superconducting order
parameters. Here we use resonant ultrasound spectroscopy to measure the entire
symmetry-resolved elastic tensor of Sr$_2$RuO$_4$ through the superconducting
transition. We observe a thermodynamic discontinuity in the shear elastic
modulus $c_{66}$, requiring that the superconducting order parameter is
two-component. A two-component $p$-wave order parameter, such as $p_x+i p_y$,
naturally satisfies this requirement. As this order parameter appears to be
precluded by recent NMR experiments, we suggest that two other two-component
order parameters, namely $\left\{d_{xz},d_{yz}\right\}$ or
$\left\{d_{x^2-y^2},g_{xy(x^2-y^2)}\right\}$, are now the prime candidates for
the order parameter of Sr$_2$RuO$_4$.
