  We present a computation, within weakly-coupled thermal QCD, of the
production rate of low invariant mass ($M^2 \sim g^2 T^2$) dileptons, at
next-to-leading order (NLO) in the coupling (which is $O(g^3 e^2 T^2)$). This
involves extending the NLO calculation of the photon rate which we recently
presented to the case of small nonzero photon invariant mass. Numerical results
are discussed and tabulated forms and code are provided for inclusion in
hydrodynamical models. We find that NLO corrections can increase the dilepton
rate by up to 30-40% relative to leading order. We find that the
electromagnetic response of the plasma for real photons and for small invariant
mass but high energy dilepton pairs (e.g., $M^2 < (300\:\mathrm{MeV})^2$ but
$p_T > 1 \: \mathrm{GeV}$) are close enough that dilepton pair measurements
really can serve as Ersatz photon measurements. We also present a matching a la
Ghisoiu and Laine between our results and results at larger invariant masses.
