  Barium monohydride (BaH) is an attractive candidate for extending laser
cooling and trapping techniques to diatomic hydrides. The apparatus and
high-resolution optical spectroscopy presented here demonstrate progress toward
this goal. A cryogenic buffer-gas-cooled molecular beam of BaH was constructed
and characterized. Pulsed laser ablation into cryogenic helium buffer gas
delivers $\sim1\times10^{10}$ molecules/sr/pulse in the X$^2\Sigma^+$
($v''=0,N''=1$) state of primary interest. More than $1\times10^7$ of these
molecules per pulse enter the downstream science region with forward velocities
below 100 m/s and transverse temperature of 0.1 K. This molecular beam enabled
high-resolution optical spectra of BaH in quantum states relevant to laser
slowing and cooling. The reported measurements include hyperfine structure and
magnetic $g$ factors in the X$^2\Sigma^+$, B$^2\Sigma^+$, and A$^2\Pi_{1/2}$
states.
