Let H\G be a causal symmetric space sitting inside its complexification H-C
\G(C). Then there exist certain G-invariant Stein subdomains Xi of H-C\G(C)
. The Haar measure on H-C\G(C) gives rise to a G-invariant measure on Xi. W
ith respect to this measure one can define the Bergman space B-2(Xi) of squ
are integrable holomorphic functions on Xi. The group G acts unitarily on t
he Hilbert space B-2(Xi) by left translations in the arguments. The main re
sult of this paper is the Plancherel Theorem for B-2(Xi), i.e., the disinte
gration formula for the left regular representation into irreducibles.