We present new observations of two anomalous X-ray pulsars (AXPs), 4U 0142+61 and 1E 1048.1-5937, made in 1998 with ASCA. The energy spectrum of each of these two AXPs is found to consist of two components, a power-law and a blackbody emission from the neutron star surface. These observations, when compared with earlier ASCA observations in 1994, show remarkable stability in the intensities, spectral shapes, and pulse profiles. However, we find that the spin-down rate in 1E 1048.1-5937 is not constant. In this source, we have clearly identified three epochs with spin-down rates different from each other and from the average value. This has very strong implications for the magnetar hypothesis for AXPs. We also note that the spin-down rate and its variations in 1E 1048.1-5937 are much larger than can normally be produced by an accretion disk with a very low mass accretion rate corresponding to its low X-ray luminosity.