The MiniBooNE experiment at Fermilab reports results from a search for ${\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}}\ensuremath{\rightarrow}{\overline{\ensuremath{\nu}}}_{e}$ oscillations, using a data sample corresponding to $5.66\ifmmode\times\else\texttimes\fi{}{10}^{20}$ protons on target. An excess of $20.9\ifmmode\pm\else\textpm\fi{}14.0$ events is observed in the energy range $475<{E}_{\ensuremath{\nu}}^{\mathrm{QE}}<1250\text{ }\text{ }\mathrm{MeV}$, which, when constrained by the observed ${\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}}$ events, has a probability for consistency with the background-only hypothesis of 0.5%. On the other hand, fitting for ${\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}}\ensuremath{\rightarrow}{\overline{\ensuremath{\nu}}}_{e}$ oscillations, the best-fit point has a ${\ensuremath{\chi}}^{2}$ probability of 8.7%. The data are consistent with ${\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}}\ensuremath{\rightarrow}{\overline{\ensuremath{\nu}}}_{e}$ oscillations in the 0.1 to $1.0\text{ }\text{ }{\mathrm{eV}}^{2}$ $\ensuremath{\Delta}{m}^{2}$ range and with the evidence for antineutrino oscillations from the Liquid Scintillator Neutrino Detector at Los Alamos National Laboratory.