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We find all irreducible constituents of the Weil representation of a unitarygroup $U_m(A)$ of rank $m$ associated to a ramified quadratic extension $A$ ofa finite, commutative, local and principal ring $R$ of odd characteristic. Weshow that this Weil representation is multiplicity free with monomialirreducible constituents. We also find the number of these constituents anddescribe them in terms of Clifford theory with respect to a congruencesubgroup. We find all character degrees in the special case when $R$ is afield.

The Weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring