  We have used infrared and radio observations to search for a dwarf galaxy
associated with the high-velocity cloud (HVC) known as Complex H. Complex H is
a large (> 400 deg^2) and probably nearby (d = 27 kpc) HVC whose location in
the Galactic plane has hampered previous investigations of its stellar content.
The HI mass of the cloud is 2.0 x 10^7 (d/27 kpc)^2 Msun, making Complex H one
of the most massive HVCs if its distance is > ~20 kpc. Virtually all similar HI
clouds in other galaxy groups are associated with LSB dwarf galaxies. We
selected mid-IR MSX sources in the direction of Complex H that appeared likely
to be star-forming regions and observed them at the wavelength of the CO 1-0
transition. 59 of the 60 observed sources show emission at Milky Way
velocities, and we detected no emission at velocities consistent with that of
Complex H. We use these observations to set an upper limit on the ongoing star
formation rate in the HVC of < 5 x 10^-4 Msun yr^-1. We also searched the 2MASS
database for evidence of any dwarf-galaxy-like stellar population in the
direction of the HVC and found no trace of a distant red giant population, with
an upper limit on the stellar mass of ~10^6 Msun. Given the lack of evidence
for either current star formation or an evolved population, we conclude that
Complex H cannot be a dwarf galaxy with properties similar to those of known
dwarfs. Complex H is therefore one of the most massive known HI clouds without
stars. If Complex H is self-gravitating, then it is one of the few known dark
galaxy candidates. These findings may offer observational support for the idea
that the Cold Dark Matter substructure problem is related to the difficulty of
forming stars in low-mass dark matter halos; alternatively, Complex H could be
an example of a cold accretion flow onto the Milky Way. [slightly abridged]
