  The photon-absorption edge in a weakly interacting one-dimensional electron
gas is studied, treating backscattering of conduction electrons from the core
hole exactly. Close to threshold, there is a power-law singularity in the
absorption, $I(\epsilon) \propto \epsilon^{-\alpha}$, with $\alpha = 3/8 +
\delta_+/\pi - \delta_+^2/2\pi^2$ where $\delta_+$ is the forward scattering
phase shift of the core hole. In contrast to previous theories, $\alpha$ is
finite (and universal) in the limit of weak core hole potential. In the case of
weak backscattering $U(2k_F)$, the exponent in the power-law dependence of
absorption on energy crosses over to a value $\alpha = \delta_+/\pi -
\delta_+^2/2\pi^2$ above an energy scale $\epsilon^* \sim
[U(2k_F)]^{1/\gamma}$, where $\gamma$ is a dimensionless measure of the
electron-electron interactions.
