  We study phase diffusion in a Bose-Einstein condensate of light in a
dye-filled optical microcavity, i.e., the spreading of the probability
distribution for the condensate phase. To observe this phenomenon, we propose
an interference experiment between the condensed photons and an external laser.
We determine the average interference patterns, considering quantum and thermal
fluctuations as well as dissipative effects due to the dye. Moreover, we show
that a representative outcome of individual measurements can be obtained from a
stochastic equation for the global phase of the condensate.
