  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, possibly, perturbations very small as
$\Vert x \Vert \to \infty$. By proving both generation and motion of interface
properties, we show that the sharp interface limit moves by a constant speed,
which is the minimal speed of some related one-dimensional travelling waves. We
obtain an estimate of the thickness of the transition layers. We also exhibit
initial data "not so small" at infinity which do not allow the interface
phenomena.
