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DNA strand displacement, where a single-stranded nucleic acid invades a DNA duplex, is pervasive in genomic processes and DNA engineering applications. The kinetics of strand displacement have been studied in bulk; however, the kinetics of the underlying strand exchange were obfuscated by a slow bimolecular association step. Here, we use a novel single-molecule Fluorescence Resonance Energy Transfer (smFRET) approach termed the "fission" assay to obtain the full distribution of first passage times of unimolecular strand displacement. At a frame time of 4.4 ms, the first passage time distribution for a 14-nt displacement domain exhibited a nearly monotonic decay with little delay. Among the eight different sequences we tested, the mean displacement time was on average 35 ms and varied by up to a factor of 13. The measured displacement kinetics also varied between complementary invaders and between RNA and DNA invaders of the same base sequence except for T$\rightarrow$U substitution. However, displacement times were largely insensitive to the monovalent salt concentration in the range of 0.25 M to 1 M. Using a one-dimensional random walk model, we infer that the single-step displacement time is in the range of $\sim 30 μs$ to $\sim 300 μs$ depending on the base identity. The framework presented here is broadly applicable to the kinetic analysis of multistep processes investigated at the single-molecule level.