Evidence is presented for the associated production of a single top quark and $W$ boson in $pp$ collisions at $\sqrt{s}=7\text{ }\text{ }\mathrm{TeV}$ with the CMS experiment at the LHC. The analyzed data correspond to an integrated luminosity of $4.9\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. The measurement is performed using events with two leptons and a jet originated from a $b$ quark. A multivariate analysis based on kinematic properties is utilized to separate the $t\overline{t}$ background from the signal. The observed signal has a significance of $4.0\ensuremath{\sigma}$ and corresponds to a cross section of ${16}_{\ensuremath{-}4}^{+5}\text{ }\text{ }\mathrm{pb}$, in agreement with the standard model expectation of $15.6\ifmmode\pm\else\textpm\fi{}{0.4}_{\ensuremath{-}1.2}^{+1.0}\text{ }\text{ }\mathrm{pb}$.