We present the first results of an observational programme at the ESO Very Large Telescope aimed at detecting a large sample of high-redshift galaxies fainter than the current spectroscopic limit of R=25.5 for Lyman-Break galaxies. In this paper, we describe the results of deep narrow and broad-band imaging and subsequent follow-up multi-object spectroscopy of faint high-redshift galaxies in the fields of the BRI1346-0322 and Q2138-4427. These QSOs have intervening absorbers, at redshifts z=2.85 and z=3.15 respectively, for which redshifted Ly-alpha emission falls within less than a few AA from the central wavelengths of existing VLT (~60 AA wide) narrow-band filters. We selected 37 and 27 candidate emission-line galaxies in the two fields respectively. About 85% of the candidates have R-band magnitudes fainter than R=25.5. The first spectroscopic follow-up of a sub-sample of the candidates resulted in 41 confirmed candidates and 4 foreground galaxies (three [OII] emitters and one CIV emitter). The confirmation rate is 82% and 68% in the field of BRI1346-0322 and Q2138-4427 respectively. In addition, we serendipitously detect a number of other emission-line sources on some of the slitlets not used for candidates. Of these, 9 are also most likely Ly-alpha emitters with redshifts ranging from 1.98 to 3.47. The redshift distribution of confirmed candidates in the field of BRI1346-0322 is consistent with being drawn from a uniform distribution weighted by the filter response curve, whereas the galaxies in the field of Q2138-4427 have redshifts clustering very close to the redshift of the damped Ly-alpha absorber. This latter fact indicates the existence of a large `pancake-like structure confirming the earlier suggestions of Francis & Hewitt (1993).
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