  In this article, we consider the Parabolic Anderson Model with constant
initial condition, driven by a space-time homogeneous Gaussian noise, with
general covariance function in time and spatial spectral measure satisfying
Dalang's condition. First, we prove that the solution (in the Skorohod sense)
exists and is continuous in $L^p(\Omega)$. Then, we show that the solution has
a modification whose sample paths are H\"older continuous in space and time,
with optimal exponents, and under the minimal condition on the spatial spectral
measure of the noise (which is the same as the condition encountered in the
case of the white noise in time). This improves similar results which were
obtained in Hu, Huang, Nualart and Tindel (2015), and Song (2017) under more
restrictive conditions, and with sub-optimal exponents for H\"older continuity.
