  We find all irreducible constituents of the Weil representation of a unitary
group $U_m(A)$ of rank $m$ associated to a ramified quadratic extension $A$ of
a finite, commutative, local and principal ring $R$ of odd characteristic. We
show that this Weil representation is multiplicity free with monomial
irreducible constituents. We also find the number of these constituents and
describe them in terms of Clifford theory with respect to a congruence
subgroup. We find all character degrees in the special case when $R$ is a
field.
