We report on a method developed for characterization the third order optical nonlinearity of singlemode polymer optical fiber. The Sagnac homodyning method yields values for both the real and imaginary components of (chi) (3) from a single measurement. The method also allows for the separation of the effects due to nonlinear mechanisms having response times on the order of milliseconds or slower. The 10micrometers core of the characterized fiber was doped with 0.15 percent by weight of the squaraine HSQ. The real component of the nonlinearity was determined to be Re((chi) (3) equals -9.8 +/- 2.5 X 10-12 esu, and the imaginary component Im((chi) (3) equals 5.5 +/- 0.3 X 10-12 esu.