  New bounds are proven on the mean vertical convective heat transport,
$\overline{\langle wT \rangle}$, for uniform internally heated (IH) convection
in the limit of infinite Prandtl number. For fluid in a horizontally-periodic
layer between isothermal boundaries, we show that $\overline{\langle wT
\rangle} \leq \frac12 - c R^{-2}$, where $R$ is a nondimensional `flux'
Rayleigh number quantifying the strength of internal heating and $c = 216$.
Then, $\overline{\langle wT \rangle} = 0$ corresponds to vertical heat
transport by conduction alone, while $\overline{\langle wT \rangle} > 0$
represents the enhancement of vertical heat transport upwards due to convective
motion. If, instead, the lower boundary is a thermal insulator, then we obtain
$\overline{\langle wT \rangle} \leq \frac12 - c R^{-4}$, with $c\approx
0.0107$. This result implies that the Nusselt number $Nu$, defined as the ratio
of the total-to-conductive heat transport, satisfies $Nu \lesssim R^{4}$. Both
bounds are obtained by combining the background method with a minimum principle
for the fluid's temperature and with Hardy--Rellich inequalities to exploit the
link between the vertical velocity and temperature. In both cases, power-law
dependence on $R$ improves the previously best-known bounds, which, although
valid at both infinite and finite Prandtl numbers, approach the uniform bound
exponentially with $R$.
