  We report on the 2004 June 29 burst detected from the direction of the
Anomalous X-ray Pulsar (AXP) 1E 1048.1-5937 using the Rossi X-ray Timing
Explorer (RXTE). We find a simultaneous increase of ~3.5 times the quiescent
value in the 2-10 keV pulsed flux of 1E 1048.1-5937 during the tail of the
burst which identifies the AXP as the burst's origin. The burst was overall
very similar to the two others reported from the direction of this source in
2001. The unambiguous identification of 1E 1048.1-5937 as the burster here
confirms it was the origin of the 2001 bursts as well. The epoch of the burst
peak was very close to the arrival time of 1E 1048.1-5937's pulse peak. The
burst exhibited significant spectral evolution with the trend going from hard
to soft. During the 11 days following the burst, the AXP was observed further
with RXTE, XMM-Newton and Chandra. Pre- and post-burst observations revealed no
change in the total flux or spectrum of the quiescent emission. Comparing all
three bursts detected thus far from this source we find that this event was the
most fluent (>3.3x10^-8 erg/cm^2 in the 2-20 keV band), had the highest peak
flux (59+/-9x10^-10 erg/s/cm^2 in the 2-20 keV band), and the longest duration
(>699 s). The long duration of the burst differentiates it from Soft Gamma
Repeater (SGR) bursts which have typical durations of ~0.1 s. Bursts that occur
preferentially at pulse maximum, have fast-rises and long X-tails containing
the majority of the total burst energy have been seen uniquely from AXPs. The
marked differences between AXP and SGRs bursts may provide new clues to help
understand the physical differences between these objects.
