  Results of Struwe, Grillakis, Struwe-Shatah, Kapitanski, Bahouri-Shatah,
Bahouri-G\'erard and Nakanishi have established global wellposedness,
regularity, and scattering in the energy class for the energy-critical
nonlinear wave equation $\Box u = u^5$ in $\R^{1+3}$, together with a spacetime
bound $$ \| u \|_{L^4_t L^{12}_x(\R^{1+3})} \leq M(E(u))$$ for some finite
quantity M(E(u)) depending only on the energy E(u) of u. We reprove this
result, and show that this quantity obeys a bound of at most exponential type
in the energy, and specifically $M(E) \leq C (1+E)^{C E^{105/2}}$ for some
absolute constant C > 0. The argument combines the quantitative local potential
energy decay estimates of these previous papers with arguments used by Bourgain
and the author for the analogous nonlinear Schr\"odinger equation.
