If I remember correctly, it is isomorphic to the group of ordinary (virtual) characters over $K=\mbox{Frac}(\mathcal{O})$, the field of fractions of $\mathcal{O}$. That is, the Grothendieck group of $KG$. An isomorphism is given by simply tensoring with $K$.
This was proved by Swan in 'The Grothendieck ring of a finite group', Topology 2, 85-110, 1963. He proves a more general result over an arbitrary integral domain there, and Theorem 3 is the one you're after.