  The classical construction of the Weil representation, with complex
coefficients, has long been expected to work for more general coefficient
rings. This paper exhibits the minimal ring $\mathcal{A}$ for which this is
possible, the integral closure of $\mathbb{Z}[\frac{1}{p}]$ in a cyclotomic
field, and carries out the construction of the Weil representation over
$\mathcal{A}$-algebras. As a leitmotif all along the work, most of the problems
can actually be solved over the base ring $\mathcal{A}$ and transferred to any
$\mathcal{A}$-algebra by scalar extension. The most striking fact is that all
these Weil representations arise as the scalar extension of a single one with
coefficients in $\mathcal{A}$. In this sense, the Weil module obtained is
universal. Building upon this construction, we speculate and make predictions
about an integral theta correspondence.
