Abstract: We derive an improved rigorous lower bound on the space and time averaged temperature $\langle T\rangle$ of an infinite Prandtl number Boussinesq fluid contained between isothermal no-slip boundaries thermally driven by uniform internal heating. A novel approach is used wherein a singular stable stratification is introduced as a perturbation to a non-singular background profile, yielding the estimate $\langle T\rangle \geq 0.419 \left[R \, \log R\right]^{-1/4}$ where $R$ is the heat Rayleigh number. The analysis relies on a generalized Hardy-Rellich inequality that is proved via a new approach, ensuring the strictness of the inequality.