We used a multiplexed approach based on flow-stretched DNA to monitor the enzymatic digestion of lambda-phage DNA by individual bacteriophage lambda exonuclease molecules. Statistical analyses of multiple single-molecule trajectories observed simultaneously reveal that the catalytic rate is dependent on the local base content of the substrate DNA. By relating single-molecule kinetics to the free energies of hydrogen bonding and base stacking, we establish that the melting of a base from the DNA is the rate-limiting step in the catalytic cycle. The catalytic rate also exhibits large fluctuations independent of the sequence, which we attribute to conformational changes of the enzyme-DNA complex.