  We present observations of the dark Gamma-Ray Burst GRB 051008 provided by
Swift/BAT, Swift/XRT, Konus-WIND, INTEGRAL/SPI-ACS in the high-energy domain
and the Shajn, Swift/UVOT, Tautenburg, NOT, Gemini and Keck I telescopes in the
optical and near-infrared bands. The burst was detected only in gamma- and
X-rays and neither a prompt optical nor a radio afterglow were detected down to
deep limits. We identified the host galaxy of the burst, which is a typical
Lyman-break Galaxy (LBG) with R-magnitude of 24.06 +/- 0.10. A redshift of the
galaxy of z = 2.77 (-0.20,+0.15) is measured photometrically due to the
presence of a clear, strong Lyman-break feature. The host galaxy is a small
starburst galaxy with moderate intrinsic extinction (A_V = 0.3 mag) and has a
SFR of ~ 60 M_Sun / yr typical for LBGs. It is one of the few cases where a GRB
host has been found to be a classical Lyman-break galaxy. Using the redshift we
estimate the isotropic-equivalent radiated energy of the burst to be E_iso =
(1.15 +/- 0.20) x 10^54 erg. We also provide evidence in favour of the
hypothesis that the darkness of GRB 051008 is due to local absorption resulting
from a dense circumburst medium.
