  We show that the fractional Sobolev inequality for the embedding $\H
\hookrightarrow L^{\frac{2N}{N-s}}(\R^N)$, $s \in (0,N)$ can be sharpened by
adding a remainder term proportional to the distance to the set of optimizers.
As a corollary, we derive the existence of a remainder term in the weak
$L^{\frac{N}{N-s}}$-norm for functions supported in a domain of finite measure.
Our results generalize earlier work for the non-fractional case where $s$ is an
even integer.
