For the infinite-Prandtl-number limit of the Boussinesq equations, the enhancement of vertical heat transport in Rayleigh–Bénard convection, the Nusselt number $\hbox{\it Nu}$, is bounded above in terms of the Rayleigh number $\hbox{\it Ra}$ according to $\hbox{\it Nu}\,{\le}\,0.644 \,{\times}\hbox{\it Ra}^{{1}/{3}} [\log{\hbox{\it Ra}}]^{{1}/{3}}$ as $\hbox{\it Ra}\,{\rightarrow}\,\infty$. This result follows from the utilization of a novel logarithmic profile in the background method for producing bounds on bulk transport, together with new estimates for the bi-Laplacian in a weighted $L^{2}$ space. It is a quantitative improvement of the best currently available analytic result, and it comes within the logarithmic factor of the pure 1/3 scaling anticipated by both the classical marginally stable boundary layer argument and the most recent high-resolution numerical computations of the optimal bound on $\hbox{\it Nu}$ using the background method.