  We investigate the singular limit, as $\ep \to 0$, of the Fisher equation
$\partial_t u=\ep \Delta u + \ep ^{-1}u(1-u)$ in the whole space. We consider
initial data with compact support plus perturbations with {\it slow exponential
decay}. We prove that the sharp interface limit moves by a constant speed,
which dramatically depends on the tails of the initial data. By performing a
fine analysis of both the generation and motion of interface, we provide a new
estimate of the thickness of the transition layers.
