We prove a new rigorous upper bound on the vertical heat transport for Bénard–Marangoni convection of a two- or three-dimensional fluid layer with infinite Prandtl number. Precisely, for Marangoni number 𝑀𝑎≫1 the Nusselt number 𝑁𝑢 is bounded asymptotically by 𝑁𝑢⩽const.×𝑀𝑎2/7(ln𝑀𝑎)−1/7 . Key to our proof are a background temperature field with a hyperbolic profile near the fluid’s surface and new estimates for the coupling between temperature and vertical velocity.