Relativity requires that it have no mass if it indeed does travel at c.
Ooh... ooh... careful! Be sure you're talking about
rest mass rather than relativistic mass. Photons have no rest mass, but they still carry momentum, after all.
Every photon that leaves the Sun carries away a little bit of it's 'mass' if you like, since the photons are generated in nuclear fusion processes which release a small amount of nuclear binding energy as electromagnetic energy. An interesting fact is that since the Sun is largely opaque, a photon created in the core takes, on average, about a million years to get out to the surface where it can escape. The light you see from the Sun now was created by fusion a million years ago, and there's a million years' worth of fusion photons trapped inside it at all times. Cool huh?