Numerical integration involving expensive target functions is a common bottleneck in Bayesian inference and simulation. When a cheap surrogate is available, standard approaches such as reweighting or importance sampling often suffer from high variance and inefficient use of function evaluations. We introduce Slice Monte Carlo integration (S$\ell$MC), a method that leverages a Nested Sampling-like procedure on the surrogate to partition the space into informative strata, or slices, while generating samples in the parameter space drawn from the prior within each slice. This enables stratified Monte Carlo integration of the expensive target function over the surrogate-induced partition, yielding an efficient estimate of the target integral. The surrogate level sets therefore make the induced ordering of the parameter space the relevant information for S$\ell$MC, rather than the pointwise target-to-surrogate ratios governing importance-sampling efficiency. Another key advantage of S$\ell$MC is the decoupling of slice volume estimation from the evaluation of the target, allowing the refinement of the slice-volume estimates without additional target evaluations. We investigate the properties of S$\ell$MC, demonstrate how to efficiently generate posterior samples, and assess its robustness in controlled Gaussian benchmark families with continuously tunable surrogate-target mismatch.

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