  Coarsening on a one-dimensional lattice is described by the voter model or
equivalently by coalescing (or annihilating) random walks representing the
evolving boundaries between regions of constant color and by backward (in time)
coalescing random walks corresponding to color genealogies. Asympotics for
large time and space on the lattice are described via a continuum space-time
voter model whose boundary motion is expressed by the {\it Brownian web} (BW)
of coalescing forward Brownian motions. In this paper, we study how small noise
in the voter model, corresponding to the nucleation of randomly colored
regions, can be treated in the continuum limit. We present a full construction
of the continuum noisy voter model (CNVM) as a random {\it quasicoloring} of
two-dimensional space time and derive some of its properties. Our construction
is based on a Poisson marking of the {\it backward} BW within the {\it double}
(i.e., forward and backward) BW.
