  The ANTARES collaboration has performed a series of {\em in situ}
measurements to study the background light for a planned undersea neutrino
telescope. Such background can be caused by $^{40}$K decays or by biological
activity. We report on measurements at two sites in the Mediterranean Sea at
depths of 2400~m and 2700~m, respectively. Three photomultiplier tubes were
used to measure single counting rates and coincidence rates for pairs of tubes
at various distances. The background rate is seen to consist of three
components: a constant rate due to $^{40}$K decays, a continuum rate that
varies on a time scale of several hours simultaneously over distances up to at
least 40~m, and random bursts a few seconds long that are only correlated in
time over distances of the order of a meter. A trigger requiring coincidences
between nearby photomultiplier tubes should reduce the trigger rate for a
neutrino telescope to a manageable level with only a small loss in efficiency.
