  This paper is concerned with a priori $C^\infty$ regularity for
three-dimensional doubly periodic travelling gravity waves whose fundamental
domain is a symmetric diamond. The existence of such waves was a long standing
open problem solved recently by Iooss and Plotnikov. The main difficulty is
that, unlike conventional free boundary problems, the reduced boundary system
is not elliptic for three-dimensional pure gravity waves, which leads to small
divisors problems. Our main result asserts that sufficiently smooth diamond
waves which satisfy a diophantine condition are automatically $C^\infty$. In
particular, we prove that the solutions defined by Iooss and Plotnikov are
$C^\infty$. Two notable technical aspects are that (i) no smallness condition
is required and (ii) we obtain an exact paralinearization formula for the
Dirichlet to Neumann operator.
