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 40K 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 40K 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.Comment: 18 pages, 8 figures, accepted for publication in Astroparticle
Physic