Threshold photodetachment spectroscopy of the positronium negative ion has been accomplished for the first time employing an efficient source of the ions and photodetachment techniques combined with a tunable optical parametric oscillator and amplifier laser. The photodetachment threshold, corresponding to the electron affinity of positronium (1(3)S(1)), was determined to be 326.88 +/- 0.09(stat) +/- 0.10(syst) meV by laser photodetachment threshold measurements. This result is consistent with a variational calculation corrected for leading relativistic and quantum electrodynamical effects.