Using X-ray data from the Rossi X-Ray Timing Explorer (RXTE), we carried out a pulse-timing analysis of the transient X-ray pulsar SAX J2103.5+4545. An outburst was detected by the All-Sky Monitor (ASM) in 1999 October 25 and reached a peak X-ray brightness of 27 mcrab on October 28. Between November 19 and December 27, the RXTE Proportional Counter Array (PCA) carried out pointed observations that provided us with pulse arrival times. These yield an eccentric orbit (e = 0.4 +/- 0.2) with an orbital period of 12.68 +/- 0.25 days and a light-travel time across the projected semimajor axis of 72 +/- 6 s. The pulse period was measured to be 358.62171 +/- 0.00088 s, and the spin-up rate was (2.50 +/- 0.15) x 10(-13) Hz s(-1). The ASM data for the 1997 February-September outburst, in which F. Hulleman, J. J. M. in 't Zand, & J. Heise discovered SAX J2103.5 + 4545 using BeppoSAX, are modulated at timescales close to the orbital period. Folded light curves of the 1997 ASM data and the 1999 PCA data are similar and show that the intensity increases at periastron passages.