  We present a consistent theory of N=1 supergravity in twelve-dimensions with
the signature (10,2). Even though the formulation uses two null vectors
violating the manifest Lorentz covariance, all the superspace Bianchi
identities are satisfied. After a simple dimensional reduction to
ten-dimensions, this theory reproduces the N=1 supergravity in ten-dimensions,
supporting the consistency of the system. We also show that our supergravity
can be the consistent backgrounds for heterotic or type-I superstring in
Green-Schwarz formulation, by confirming the kappa-invariance of the total
action. This theory is supposed to be the purely N=1 supergravity sector for
the field theory limit of the recently predicted F-theory in twelve-dimensions.
