  We study the spectra of general $N\times N$ Toeplitz matrices given by
symbols in the Wiener Algebra perturbed by small complex Gaussian random
matrices, in the regime $N\gg 1$. We prove an asymptotic formula for the number
of eigenvalues of the perturbed matrix in smooth domains. We show that these
eigenvalues follow a Weyl law with probability sub-exponentially close to $1$,
as $N\gg1$, in particular that most eigenvalues of the perturbed Toeplitz
matrix are close to the curve in the complex plane given by the symbol of the
unperturbed Toeplitz matrix.
