We introduce sweeping capillary electrophoresis (SweepCE), a non-stopped-flow method for directly measuring the bimolecular rate constant of complex formation, and demonstrate its use for studying protein-DNA interaction. The capillary is prefilled with a solution of DNA, and electrophoresis is then carried out from a solution of the protein in a continuous mode. Because the electrophoretic mobility of the protein is greater than that of DNA, the protein continuously mixes with DNA and forms the protein-DNA complex. The complex migrates with a velocity higher than that of DNA and causes sweeping of DNA, which gave the name to the method. The bimolecular rate constant, kon, of complex formation can be determined from the time profile of DNA concentration using a simple mathematical model of the sweeping process. In this proof-of-principle work, we used SweepCE to directly measure kon = (3.4 +/- 0.6) x 106 M-1 s-1 for the interaction between single-stranded DNA-binding protein and a 15-mer DNA oligonucleotide. Along with nonequilibrium capillary electrophoresis of equilibrium mixtures (NECEEM), SweepCE establishes a universal and comprehensive platform for studying kinetic and equilibrium parameters of complex formation between biopolymers.