The generalized Riemann integral of Pfeffer (1991) is defined on all bounded $\mathrm{BV}$ subsets of ${ℝ}^n$, but it is additive only with respect to pairs of disjoint sets whose closures intersect in a set of $\sigma$-finite Hausdorff measure of codimension one. Imposing a stronger regularity condition on partitions of $\mathrm{BV}$ sets, we define a Riemann-type integral which satisfies the usual additivity condition and extends the integral of Pfeffer. The new integral is lipeomorphism-invariant and closed with respect to the formation...