  The astrophysical capture process $\alpha+d$ $\rightarrow$ $^6$Li + $\gamma$
is studied in a three-body model.
  The initial state is factorized into the deuteron bound state and the
$\alpha+d$ scattering state. The final nucleus $^6$Li(1+) is described as a
three-body bound state $\alpha+n+p$ in the hyperspherical Lagrange-mesh method.
  The contribution of the E1 transition operator from the initial isosinglet
states to the isotriplet components of the final state is estimated to be
negligible. An estimation of the forbidden E1 transition to the isosinglet
components of the final state is comparable with the corresponding results of
the two-body model.
  However, the contribution of the E2 transition operator is found to be much
smaller than the corresponding estimations of the two-body model.
  The three-body model perfectly matches the new experimental data of the LUNA
collaboration with the spectroscopic factor 2.586 estimated from the
bound-state wave functions of $^6$Li and deuteron.
