  Given an infinite set of special divisors satisfying a mild regularity
condition, we prove the existence of a Borcherds product of non-zero weight
whose divisor is supported on these special divisors. We also show that every
meromorphic Borcherds product is the quotient of two holomorphic ones. The
proofs of both results rely on the properties of vector valued Eisenstein
series for the Weil representation.
