  The non-relativistic three-body Schr\"odinger equation of the heteronuclear
molecular ion HD$^+$ is solved in perimetric coordinates using the
Lagrange-mesh method. Energies and wave functions of the four lowest
vibrational bound or quasibound states $v=0-3$ are calculated for total orbital
momenta from 0 to 47. Energies are given with an accuracy from about 12 digits
for the lowest vibrational level to at least 9 digits for the third vibrational
excited level. With a simple calculation using the corresponding wave
functions, accurate dipole transition probabilities per time unit between those
levels are given over the whole $v=0-3$ rotational bands. Results are presented
with six significant figures.
