The first observation of the decay K^+ --> pi^+ gamma gamma is reported. A total of 31 events was observed with an estimated background of 5.1 +- 3.3 events in the pi+ momentum range from 100 MeV/c to 180 MeV/c. The corresponding partial branching ratio, B(K+ -> pi+ gamma gamma, 100 MeV/c < P_pi^+ < 180 MeV/c), is (6.0 +- 1.5 (stat) +- 0.7 (sys)) x 10^{-7}. No K^+ --> pi^+ gamma gamma decay was observed in the pi^+ momentum region greater than 215 MeV/c. The observed pi^+ momentum spectrum is compared with the predictions of chiral perturbation theory.