  We prove a new rigorous bound for the mean vertical heat transport $\langle w
T \rangle$ in internally heated convection with an insulating lower boundary
and an upper boundary with fixed heat flux that balances the total internal
heating. An analytical application of the background method, based on the
construction of a quadratic auxiliary function, yields $\langle w T \rangle
\leq \tfrac{1}{2}(\tfrac{1}{2}+ \tfrac{1}{\sqrt{3}} ) - 1.6899\, R^{-1/3}$,
where $R$ is the Rayleigh number. Numerical optimisation of the auxiliary
function suggests that the asymptotic value of this bound and the $-1/3$
exponent are optimal within our bounding framework. This new result halves the
best existing (uniform) bound and the Rayleigh number dependence is consistent
with previous conjectures and heuristic scaling arguments.
