We present new X-ray observations of the X-ray pulsar 1E 2259+586, obtained during March 1997, with the Rossi X-Ray Timing Explorer (RXTE). We have measured the pulse frequency derivative (nu) over dot = (-1.08 +/- 0.04) x 10(-14) Hz s(-1) from pulse arrival times obtained in a sequence of 5 observations spread over one month. This (nu) over dot is consistent with the long term spin-down trend. We also found that the observed X-ray luminosity is consistent with that measured at quiescent X-ray flux levels by previous missions. Our observations imply that 1E 2259 +/- 586 was spinning down steadily without exhibiting any stochastic torque noise fluctuations during the month covered by our observations.