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 LX ~ 4 × 1034 erg s–1) 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 nonthermal 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 NS core, such outbursts should recur every ~10 yr.