  In this article, we study the Parabolic Anderson Model driven by a space-time
homogeneous Gaussian noise on $\mathbb{R}_{+} \times \mathbb{R}^d$, whose
covariance kernels in space and time are locally integrable non-negative
functions, which are non-negative definite (in the sense of distributions). We
assume that the initial condition is given by a signed Borel measure on
$\mathbb{R}^d$, and the spectral measure of the noise satisfies Dalang's (1999)
condition. Under these conditions, we prove that this equation has a unique
solution, and we investigate the magnitude of the $p$-th moments of the
solution, for any $p \geq 2$. In addition, we show that this solution has a
H\"older continuous modification with the same regularity and under the same
condition as in the case of the white noise in time, regardless of the temporal
covariance function of the noise.
