  We present deep spectroscopic measurements of 18 distant field galaxies
identified as gravitationally-lensed arcs in a Hubble Space Telescope image of
the cluster Abell 2218. Redshifts of these objects were predicted by Kneib et
al. (1996) using a lensing analysis constrained by the properties of two bright
arcs of known redshift and other multiply-imaged sources. The new spectroscopic
identifications were obtained using long exposures with the LDSS-2 spectrograph
on the William Herschel Telescope and demonstrate the capability of that
instrument to new limits, $R\simeq$24; the lensing magnification implies true
source magnitudes as faint as $R\simeq$25. Statistically, our measured
redshifts are in excellent agreement with those predicted from Kneib et al.'s
lensing analysis which gives considerable support to the redshift distribution
derived by the lensing inversion method for the more numerous and fainter
arclets extending to $R\simeq 25.5$. We explore the remaining uncertainties
arising from both the mass distribution in the central regions of Abell 2218
and the inversion method itself, and conclude that the mean redshift of the
faint field population at $R\simeq$25.5 ($B\sim 26$-27) is low, $<z>$=0.8-1. We
discuss this result in the context of redshift distributions estimated from
multi-colour photometry. Although such comparisons are not straightforward, we
suggest that photometric techniques may achieve a reasonable level of agreement
particularly when they include near-infrared photometry with discriminatory
capabilities in the $1<z<$2 range.

Astrophysics