  This paper extends, in a sharp way, the famous Efimov's Theorem to immersed
ends in $\real^3$. More precisely, let $M$ be a non-compact connected surface
with compact boundary. Then there is no complete isometric immersion of $M$
into $\Bbb R^3$ satisfying that $\int_M |K|=+\infty$ and $K\le-\kappa<0$, where
$\kappa$ is a positive constant and $K$ is the Gaussian curvature of $M$. In
particular Efimov's Theorem holds for complete Hadamard immersed surfaces,
whose Gaussian curvature $K$ is bounded away from zero outside a compact set.
