New bounds are proven on the mean vertical convective heat transport, ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯, 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 ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯≤12−cR−2, where R is a nondimensional `flux' Rayleigh number quantifying the strength of internal heating and c=216. Then, ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯=0 corresponds to vertical heat transport by conduction alone, while ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯>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 ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯≤12−cR−4, with c≈0.0107. This result implies that the Nusselt number Nu, defined as the ratio of the total-to-conductive heat transport, satisfies Nu≲R4. 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.