  We develop a quantum Monte Carlo procedure, in the valence bond basis, to
measure the Renyi entanglement entropy of a many-body ground state as the
expectation value of a unitary {\it Swap} operator acting on two copies of the
system. An improved estimator involving the ratio of {\it Swap} operators for
different subregions enables simulations to converge the entropy in a time
polynomial in the system size. We demonstrate convergence of the Renyi entropy
to exact results for a Heisenberg chain. Finally, we calculate the scaling of
the Renyi entropy in the two-dimensional Heisenberg model and confirm that the
N\'eel groundstate obeys the expected area law for systems up to linear size
L=28.
