  In this paper we show that any sequence of infinite lattice constellations
which is good for the unconstrained Gaussian channel can be shaped into a
capacity-achieving sequence of codes for the power-constrained Gaussian channel
under lattice decoding and non-uniform signalling. Unlike previous results in
the literature, our scheme holds with no extra condition on the lattices (e.g.
quantization-goodness or vanishing flatness factor), thus establishing a direct
implication between AWGN-goodness, in the sense of Poltyrev, and
capacity-achieving codes. Our analysis uses properties of the discrete Gaussian
distribution in order to obtain precise bounds on the probability of error and
achievable rates. In particular, we obtain a simple characterization of the
finite-blocklength behavior of the scheme, showing that it approaches the
optimal dispersion coefficient for \textit{high} signal-to-noise ratio. We
further show that for \textit{low} signal-to-noise ratio the discrete Gaussian
over centered lattice constellations can not achieve capacity, and thus a shift
(or "dither") is essentially necessary.
