  We study the behavior of the solution of a stochastic Allen-Cahn equation
$\frac{\partial u_\eps }{\partial t}=\frac 12 \frac{\partial^2 u_\eps
}{\partial x^2}-( u_\eps^3- u_\eps)+\sqrt\eps\, \dot W$, with Dirichlet
boundary conditions on a suitably large space interval $[-L_\eps , L_\eps]$,
starting from the identically zero function, and where $\dot W$ is a space-time
white noise. Our main goal is the description, in the small noise limit, of the
onset of the phase separation, with the emergence of spatial regions where
$u_\eps$ becomes close to one or another of $\pm 1$. The time scale and the
spatial structure are determined by a suitable Gaussian process that appears as
the random counterpart of the linearized A-C equation. This issue has been
initially examined by De Masi et al. [Ann. Probab. {\bf 22}, (1994), 334-371]
in the related context of a class of reaction-diffusion models obtained as a
superposition of a speeded up stirring process and a spin flip dynamics on
$\{-1,1\}^{\mathbb{Z}_\eps}$, where $\mathbb{Z}_\eps=\mathbb{Z}$ modulo
$\lfloor\eps^{-1}L_\eps\rfloor$.
