  We report the discovery of phase shifts between X-ray pulses at different
energies in the newly discovered millisecond (ms) X-ray pulsar SAX
J1808.4-3658. The results show that low-energy pulses lag high-energy pulses by
as much as $\sim$0.2 ms (or $\sim$8% of the pulse period). The measurements
were made in two different ways: (1) computing cross power spectra between
different energy bands, and (2) cross-correlating the folded pulse profiles in
different energy bands; consistent results were obtained. We speculate that the
observed soft lags might be related to the lateral expansion and subsequent
cooling of a ``hot spot'' on the neutron star surface in which the pulsed X-ray
emission originates. Also presented is the possibility of producing soft lags
via Compton down scattering of hard X-ray photons from the hot spot in the cool
surrounding atmosphere. We will discuss possible X-ray production mechanisms
for SAX J1808.4-3658 and constraints on the emission environment, based on the
observed soft lags, pulse profiles, and energy spectrum.
