  We report and study the outburst of a new transient X-ray binary (XRB) in
Terzan 5, the third detected in this globular cluster, Swift J174805.3-244637
or Terzan 5 X-3. We find clear spectral hardening in Swift/XRT data during the
outburst rise to the hard state, thanks to our early coverage (starting at L_X
~ 4x10^{34} ergs/s) of the outburst. This hardening appears to be due to the
decline in relative strength of a soft thermal component from the surface of
the neutron star (NS) during the rise. We identify a {Type I X-ray burst} in
Swift/XRT data with a long (16 s) decay time, indicative of {hydrogen burning
on the surface of the} NS. We use Swift/BAT, Maxi/GSC, Chandra/ACIS, and
Swift/XRT data to study the spectral changes during the outburst, identifying a
clear hard-to-soft state transition. We use a Chandra/ACIS observation during
outburst to identify the transient's position. Seven archival Chandra/ACIS
observations show evidence for variations in Terzan 5 X-3's non-thermal
component, but not the thermal component, during quiescence. The inferred
long-term time-averaged mass accretion rate, from the quiescent thermal
luminosity, suggests that if this outburst is typical and only slow cooling
processes are active in the neutron star core, such outbursts should recur
every ~10 years.
