We report strong instantaneous photoinduced absorption (PA) in the quasi-one-dimensional Mott insulator ${rm Sr_2CuO_3}$ in the IR spectral region. The observed PA is to an even-parity two-photon state that occurs immediately above the absorption edge. Theoretical calculations based on a two-band extended Hubbard model explains the experimental features and indicates that the strong two-photon absorption is due to a very large dipole-coupling between nearly degenerate one- and two-photon states. Room temperature picosecond recovery of the optical transparency suggests the strong potential of ${rm Sr_2CuO_3}$ for all-optical switching.