A first measurement of the polarization of $W$ bosons with large transverse momenta in $pp$ collisions is presented. The measurement is based on $36\text{ }\text{ }{\mathrm{pb}}^{\ensuremath{-}1}$ of data recorded at $\sqrt{s}=7\text{ }\text{ }\mathrm{TeV}$ by the CMS detector at the LHC. The left-handed, right-handed, and longitudinal polarization fractions (${f}_{L}$, ${f}_{R}$, and ${f}_{0}$, respectively) of $W$ bosons with transverse momenta larger than 50 GeV are determined by using decays to both electrons and muons. The muon final state yields the most precise measurement: $({f}_{L}\ensuremath{-}{f}_{R}{)}^{\ensuremath{-}}=0.240\ifmmode\pm\else\textpm\fi{}0.036(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.031(\mathrm{syst})$ and ${f}_{0}^{\ensuremath{-}}=0.183\ifmmode\pm\else\textpm\fi{}0.087(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.123(\mathrm{syst})$ for negatively charged $W$ bosons and $({f}_{L}\ensuremath{-}{f}_{R}{)}^{+}=0.310\ifmmode\pm\else\textpm\fi{}0.036(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.017(\mathrm{syst})$ and ${f}_{0}^{+}=0.171\ifmmode\pm\else\textpm\fi{}0.085(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.099(\mathrm{syst})$ for positively charged $W$ bosons. This establishes, for the first time, that $W$ bosons produced in $pp$ collisions with large transverse momenta are predominantly left-handed, as expected in the standard model.