I've heard that Deligne-Mumford's "the irreducibility..." showed this first. But I think that Mumford's "Geometric invariant theory" has its proof.
The proof is as follows: Let $H$ be the scheme that represents the functor of 3-canonical embeded smooth curves (as in proposition 5.1. of GIT). Then the moduli scheme $M$ is the geometric quotient $H/PGL$ (by proposition 5.4. of GIT). And by chapter 0, section 2 of GIT, to show the irreducibility of $M$, it's sufficient to show that of $H$. And by 5.3, $H$ is irreducible. So the moduli $M$ is also irreducible.
Is this wrong? Or, is Deligne-Mumford the first one that shows this only using algebraic tools? (GIT uses transcendental technique.)
Thank you very much!