  We construct the minimal off-shell formulation of N = 1 chiral supergravity
(SUGRA) introducing a complex antisymmetric tensor field $B_{\mu \nu}$ and a
complex axial-vector field $A_{\mu}$ as auxiliary fields. The resulting algebra
of the right- and left-handed supersymmetry (SUSY) transformations closes off
shell and generates chiral gauge transforamtions and vector gauge
transformations in addition to the transformations which appear in the case
without auxiliary fields.
