  We show that every finitely generated conical refinement monoid can be
represented as the monoid $\mathcal V(R)$ of isomorphism classes of finitely
generated projective modules over a von Neumann regular ring $R$. To this end,
we use the representation of these monoids provided by adaptable separated
graphs. Given an adaptable separated graph $(E, C)$ and a field $K$, we build a
von Neumann regular $K$-algebra $Q_K (E, C)$ and show that there is a natural
isomorphism between the separated graph monoid $M(E, C)$ and the monoid
$\mathcal V(Q_K (E, C))$.
