  We present the results of deep Lyman limit imaging in four new fields as part
of a continuing search for galaxies at $3.0 \le z \le 3.5$. The technique uses
a custom broad--band filter set ($U_n G {\cal R}$) designed to isolate objects
having Lyman continuum breaks superposed on otherwise flat--spectrum
ultraviolet continua. The observations are specifically aimed at detecting
known galaxies producing optically thick QSO metal line absorption systems, but
are equally sensitive to more generally distributed objects at the redshifts of
interest. Together with previously published results of the survey, we have now
detected in two out of six cases objects with the expected properties of
star--forming galaxies at $z \simgt 3$ within 3~arcsec of the QSO sight lines;
the two candidate absorbers have similar luminosities, $M_B \simeq -22$
($q_0=0.5$, $H_0=50$ \kms Mpc$^{-1}$), and impact parameters, $R \sim
10~h^{-1}$~kpc. The average surface density of robust Lyman break objects is
$\sim 0.5$ galaxies per square arcminute to a magnitude limit ${\cal R} =
25.0$. A simple, ``no evolution'' model based on the properties of normal
galaxies at $z \simlt 1$ predicts a density of Lyman break objects only a few
times larger than observed. We conclude that there is a substantial population
of star--forming galaxies, of relatively normal luminosity, already in place at
$z = 3 - 3.5$.
