We combine searches by the CDF and D0 Collaborations for the associated production of a Higgs boson with a $W$ or $Z$ boson and subsequent decay of the Higgs boson to a bottom-antibottom quark pair. The data, originating from Fermilab Tevatron $p\overline{p}$ collisions at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$, correspond to integrated luminosities of up to $9.7\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. The searches are conducted for a Higgs boson with mass in the range $100--150\text{ }\text{ }\mathrm{GeV}/{c}^{2}$. We observe an excess of events in the data compared with the background predictions, which is most significant in the mass range between 120 and $135\text{ }\text{ }\mathrm{GeV}/{c}^{2}$. The largest local significance is 3.3 standard deviations, corresponding to a global significance of 3.1 standard deviations. We interpret this as evidence for the presence of a new particle consistent with the standard model Higgs boson, which is produced in association with a weak vector boson and decays to a bottom-antibottom quark pair.