  We consider Schr\"odinger operators in $\ell^2(\mathbb{Z})$ whose potentials
are given by the sum of an ergodic term and a random term of Anderson type.
Under the assumption that the ergodic term is generated by a homeomorphism of a
connected compact metric space and a continuous sampling function, we show that
the almost sure spectrum arises in an explicitly described way from the
unperturbed spectrum and the topological support of the single-site
distribution. In particular, assuming that the latter is compact and contains
at least two points, this explicit description of the almost sure spectrum
shows that it will always be given by a finite union of non-degenerate compact
intervals. The result can be viewed as a far reaching generalization of the
well known formula for the spectrum of the classical Anderson model.
