We show that if $U$ is the enveloping algebra of a finite-dimensional nilpotent Lie superalgebra over a field of characteristic zero, then any graded-primitive factor ring of $U$ is isomorphic to a tensor product $C\otimes_k A$ where $C$ is the Clifford algebra of a nonsingular form over some finite field extension of $k$ and $A$ is a Weyl algebra over $k$. We prove that the same result holds for a primitive factor of $U$, except that $C$ may be either the whole Clifford algebra or just its even part. We give examples to show all possibilities can occur. Our results generalize a result of Dixmier for ordinary Lie algebras.
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