Using X-ray data from the Rossi X-ray Timing Explorer, we report the pulse timing results of the accretion-powered, high-mass X-ray binary pulsar 4U 1907 + 09, covering a time-span of almost two years. We measured three new pulse periods in addition to the previously measured four pulse periods. We are able to connect pulse arrival times in phase for more than a year. The source has been spinning down almost at a constant rate, with a spin-down rate of (nu) over dot = (- 3.54 +/- 0.02) x 10(-14) Hz s(-1) for more than 15 yr. Residuals of pulse arrival times yield a very low level of random-walk noise, with a strength of similar to2 x 10(-20) rad(2) s(-3) on a time-scale of 383 d, which is 40 times lower than that of the high-mass X-ray binary pulsar Vela X-1. The noise strength is only a factor of 5 greater than that of the low-mass X-ray binary pulsar 4U 1626-67. The low level of the timing noise and the very stable spin-down rate of 4U 1907+09 make this source unique among the high-mass X-ray binary pulsars, providing another example, in addition to 4U 1626-67, of long-term quiet spin down from an accreting source. These examples show that the extended quiet spin-down episodes observed in the anomalous X-ray pulsars 1RXS J170849.0-400910 and IE 2259+586 do not necessarily imply that these sources are not accreting pulsars.