The ANTARES collaboration has performed a series of in-situ measurements to study the background light for a planned undersea neutrino telescope. Such background can be caused by K-40 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 K-40 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.