  A numerically efficient, accurate, and easily implemented integration scheme
over convex Voronoi polyhedra (VP) is presented for use in {\it ab-initio}
electronic-structure calculations. We combine a weighted Voronoi tessellation
with isoparametric integration via Gauss-Legendre quadratures to provide
rapidly convergent VP integrals for a variety of integrands, including those
with a Coulomb singularity. We showcase the capability of our approach by first
applying to an analytic charge-density model achieving machine-precision
accuracy with expected convergence properties in milliseconds. For contrast, we
compare our results to those using shape-functions and show our approach is
greater than $10^{5}$ faster and $10^{7}$ more accurate. A weighted Voronoi
tessellation also allows for a physics-based partitioning of space that
guarantees convex, space-filling VP while reflecting accurate atomic size and
site charges, as we show within KKR methods applied to Fe-Pd alloys.
