  Weak spectral features in BL Lacertae objects (BL Lac) often provide a unique
opportunity to probe the inner region of this rare type of active galactic
nucleus. We present a Hubble Space Telescope/Cosmic Origins Spectrograph
observation of the BL Lac H 2356-309. A weak Ly$\alpha$ emission line was
detected. This is the fourth detection of a weak Ly$\alpha$ emission feature in
the ultraviolet (UV) band in the so-called "high energy peaked BL Lacs", after
Stocke et al. Assuming the line-emitting gas is located in the broad line
region (BLR) and the ionizing source is the off-axis jet emission, we constrain
the Lorentz factor ($\Gamma$) of the relativistic jet to be $\geq 8.1$ with a
maximum viewing angle of 3.6$^\circ$. The derived $\Gamma$ is somewhat larger
than previous measurements of $\Gamma \approx 3 - 5$, implying a covering
factor of $\sim$ 3% of the line-emitting gas. Alternatively, the BLR clouds
could be optically thin, in which case we constrain the BLR warm gas to be
$\sim 10^{-5}\rm\ M_{\odot}$. We also detected two HI and one OVI absorption
lines that are within $|\Delta v| < 150\rm\ km\ s^{-1}$ of the BL Lac object.
The OVI and one of the HI absorbers likely coexist due to their nearly
identical velocities. We discuss several ionization models and find a
photoionization model where the ionizing photon source is the BL Lac object can
fit the observed ion column densities with reasonable physical parameters. This
absorber can either be located in the interstellar medium of the host galaxy,
or in the BLR.
