  We report measurements of particle number correlations and fluctuations of a
photon Bose-Einstein condensate in a dye microcavity using a Hanbury
Brown-Twiss experiment. The photon gas is coupled to a reservoir of molecular
excitations, which serve both as heat bath and particle reservoir to realize
grand-canonical conditions. For large reservoirs, we observe strong number
fluctuations of order of the total particle number extending deep into the
condensed phase. Our results demonstrate that Bose-Einstein condensation under
grand-canonical ensemble conditions does not imply second-order coherence.
