  An explicit upper bound for the Weil-Petersson volumes of the moduli spaces
of punctured Riemann surfaces is obtained, using Penner's combinatorial
integration scheme with embedded trivalent graphs. It is shown that for a fixed
number of punctures n and for genus g going to infinity, the Weil-Petersson
volume of M_{g,n} has an upper bound c^g g^{2g}. Here c is an independent of n
constant, which is given explicitly.
