The order-$\ensuremath{\alpha}$ radiative correction for the $\ensuremath{\beta}$-decay process $\ensuremath{\Lambda}\ensuremath{\rightarrow}\mathrm{pe}\overline{\ensuremath{\nu}}$ is studied in order to meet the needs of high-precision experiments which measure the electron-neutrino angular distribution. The relative correction is tabulated for the decay distribution in terms of the electron energy ${E}_{e}$ and ${cos\ensuremath{\theta}}_{e\overline{\ensuremath{\nu}}}$. The one-variable distribution with respect to ${cos\ensuremath{\theta}}_{e\overline{\ensuremath{\nu}}}$ is decomposed in terms of the form-factor combinations ${f}_{1}^{2}$, ${f}_{1}{f}_{2}$, etc., and the radiative correction is evaluated for each term. For the electron-neutrino correlation parameter ${\ensuremath{\alpha}}_{e\ensuremath{\nu}}$, a huge relative correction is found due to its small value in the lowest order.