Using high-throughput tethered particle motion single-molecule experiments, the double-stranded DNA persistence length, L-p, is measured in solutions with Na+ and Mg2+ ions of various ionic strengths, I. Several theoretical equations for L-p(I) are fitted to the experimental data, but no decisive theory is found which fits all the L-p values for the two ion valencies. Properly extracted from the particle trajectory using simulations, L-p varies from 30 to 55 nm, and is comparable to previous experimental results. For the Na+-only case, L-p is an increasing concave function of I-1, well fitted by Mannings electrostatic stretching approach, but not by classical Odjik-Skolnick-Fixman theories with or without counterion condensation. With added Mg2+ ions, L-p shows a marked decrease at low I, interpreted as an ion-ion correlation effect, with an almost linear law in I-1, fitted by a proposed variational approach.