  We investigate the existence and the uniqueness of NLS ground states of fixed
mass on the half-line in the presence of a point interaction at the origin. The
nonlinearity is of power type, and the regime is either $L^2$-subcritical or
$L^{2}$-critical, while the point interaction is either attractive or
repulsive. In the $L^{2}$-subcritical case, we prove that ground states exist
for every mass value if the interaction is attractive, while ground states
exist only for sufficiently large masses if the interaction is repulsive. In
the latter case, if the power is less or equal to four, ground states coincide
with the only bound state. If instead, the power is greater than four, then
there are values of the mass for which two bound states exist, and neither of
the two is a ground state, and values of the mass for which two bound states
exist, and one of them is a ground state. In the $L^{2}$-critical case, we
prove that ground states exist for masses strictly below a critical mass value
in the attractive case, while ground states never exist in the repulsive case.
