  Let $Y$ be a smooth Enriques surface. A $K3$ carpet on $Y$ is a locally
Cohen-Macaulay double structure on $Y$ with the same invariants as a smooth
$K3$ surface (i.e., regular and with trivial canonical sheaf). The surface $Y$
possesses an \'etale $K3$ double cover $X \overset{\pi} \longrightarrow Y$. We
prove that $\pi$ can be deformed to a family $\SX \longrightarrow \mathbf
P^N_{T^*}$ of projective embeddings of $K3$ surfaces and that any projective
$K3$ carpet on $Y$ arises from such a family as the flat limit of smooth,
embedded $K3$ surfaces.
