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In this article, we consider the Parabolic Anderson Model with constantinitial condition, driven by a space-time homogeneous Gaussian noise, withgeneral covariance function in time and spatial spectral measure satisfyingDalang'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 hasa modification whose sample paths are H\"older continuous in space and time,with optimal exponents, and under the minimal condition on the spatial spectralmeasure of the noise (which is the same as the condition encountered in thecase of the white noise in time). This improves similar results which wereobtained in Hu, Huang, Nualart and Tindel (2015), and Song (2017) under morerestrictive conditions, and with sub-optimal exponents for H\"older continuity.

H\"older continuity for the Parabolic Anderson Model with space-time homogeneous Gaussian noise