The B(1/2 P-2(3/2)) --> X(1/2 (2) Sigma(+)) transition in XeBr is recorded at high resolution, using a CCD array detector to record spectra from Tesla discharge sources containing isotopically pure Xe-136 with Br-81(2) or Br-79(2). The high signal/noise capabilities of the detector permit the measurement of discrete vibrational structure in this system, which has normally been treated as a purely bound-free transition. The assignments comprise 119 upsilon’-upsilon" bands for (XeBr)-Xe-136-Br-81 and 86 for (XeBr)-Xe-136-Br-79, spanning upsilon’=0-33 and upsilon"=0-16. The van der Waals ground state is analyzed through fits to the customary polynomials in (v+1/2) and to near-dissociation expansions. Franck-Condon calculations are used to locate the X-state potential on the internuclear axis relative to the B state, which is modeled as a Rittner potential. The following fundamental spectroscopic constants (units cm(-1), for (XeBr)-Xe-136-Br-81) are obtained from the analysis : T’(e) = 35 863.2, omega’(e) = 135.72, omega(e)x(e)(’) = 0.32, omega"(e) = 25.7, omega(e)x"(e) = 0.62. The ground state has a dissociation energy D"(e) = 254 +/- 2 cm(-1) and supports 24 bound vibrational levels.