  The first metastable triplet state of HeH$^+$ was found to be present in ion
beam experiments, with its lifetime estimated to be between hundreds of
milliseconds and thousand of seconds. In this work, we use {\it ab initio}
methods to evaluate the radiative lifetimes of the six vibrational levels of
the $a\ ^3\Sigma^+$ of HeH$^+$. The transition $a\ ^3\Sigma^+ \rightarrow X \
^1\Sigma^+$ is spin-forbidden, but acquires intensity through spin-orbit
interaction with the singlet and triplet $\Pi$ states. Large scale CASSCF/MRCI
calculations using an adapted basis set were performed to determine the
potential energy curves of the relevant states of HeH$^+$ as well as the matrix
elements of the dipole and spin-orbit operators. The wave functions and
energies of the vibrational levels of the $a\ ^3\Sigma^+$ and $X \ ^1\Sigma^+$
states are obtained using a B-spline method and compared to previous works. We
find that radiative lifetime of the vibrational levels increases strongly with
$v$, the lifetime of the $v=0$ state being of 150 s. We also analyze the
contributions from discrete and continuum parts of the spectrum. With such a
long lifetime, the $a\ ^3\Sigma^+$ state could have astrophysical implications.
