  We report on the detailed study of the 2008 October outburst from the
anomalous X-ray pulsar (AXP) 1E 1547.0-5408 discovered through the Swift/Burst
Alert Telescope (BAT) detection of SGR-like short X-ray bursts on 2008 October
3. The Swift/X-ray Telescope (XRT) started observing the source after less than
100 s since the BAT trigger, when the flux (about 6E-11 erg/cm^2/s in the 2-10
keV range) was >50 times higher than its quiescent level. Swift monitored the
outbursting activity of 1E 1547.0-5408 on a daily basis for approximately three
weeks. This strategy allowed us to find a phase-coherent solution for the
source pulsations after the burst, which, besides period and period derivative,
requires a positive Period second derivative term (spin-down increase). The
time evolution of the pulse shape is complex and variable, with the pulsed
fraction increasing from 20% to 50% within the Swift observational window. The
XRT spectra can be fitted well by means of a single component, either a
power-law (PL) or a blackbody (BB). During the very initial phases of the
outburst the spectrum is hard, with a PL photon index about 2 (or kT about 1.4
keV) which steepens to about 4 (or kT about 0.8 keV) within one day from the
BAT trigger, though the two components are likely present simultaneously during
the first day spectra. An INTEGRAL observation carried out five days after the
trigger provided an upper limit of about 2E-11 erg/cm^2/s to the emission of 1E
1547.0-5408 in the 18-60 keV band.
