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 ≤ 1 2 − 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 ≤ 1 2 − 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 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.