  The beta-decay process of the $^6$He halo nucleus into the alpha+d continuum
is studied in a three-body model. The $^6$He nucleus is described as an
alpha+n+n system in hyperspherical coordinates on a Lagrange mesh. The
convergence of the Gamow-Teller matrix element requires the knowledge of wave
functions up to about 30 fm and of hypermomentum components up to K=24. The
shape and absolute values of the transition probability per time and energy
units of a recent experiment can be reproduced very well with an appropriate
alpha+d potential. A total transition probability of 1.6E-6 s$^{-1}$ is
obtained in agreement with that experiment. Halo effects are shown to be very
important because of a strong cancellation between the internal and halo
components of the matrix element, as observed in previous studies. The
forbidden bound state in the alpha+d potential is found essential to reproduce
the order of magnitude of the data. Comments are made on R-matrix fits.
