We present the first model-independent measurement of the helicity of $W$ bosons produced in top quark decays, based on a $1\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ sample of candidate $t\overline{t}$ events in the dilepton and lepton plus jets channels collected by the D0 detector at the Fermilab Tevatron $p\overline{p}$ Collider. We reconstruct the angle ${\ensuremath{\theta}}^{*}$ between the momenta of the down-type fermion and the top quark in the $W$ boson rest frame for each top quark decay. A fit of the resulting $\mathrm{cos}{\ensuremath{\theta}}^{*}$ distribution finds that the fraction of longitudinal $W$ bosons ${f}_{0}=0.425\ifmmode\pm\else\textpm\fi{}0.166(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.102(\mathrm{syst})$ and the fraction of right-handed $W$ bosons ${f}_{+}=0.119\ifmmode\pm\else\textpm\fi{}0.090(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.053(\mathrm{syst})$, which is consistent at the 30% C.L. with the standard model.