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.
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