A search has been made for a resonance in the $^{2}\mathrm{H}{(\ensuremath{\alpha},\ensuremath{\gamma})}^{6}\mathrm{Li}$ capture-reaction cross section at the energy corresponding to the ${0}^{+}$, $T=1$, 3.56-MeV state of $^{6}\mathrm{Li}$. None was found, and an upper limit of 6.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}7}$ eV was set on the parity-forbidden $\ensuremath{\alpha}$ width at the 90% confidence level. This limit is three orders of magnitude smaller than the best previous measurement.