DNA surface-hybridization biosensors utilize the selective hybridization of target sequences in solution to surface-immobilized probes. In this process, the target is usually assumed to be in excess, so that its concentration does not significantly vary while hybridizing to the surface-bound probes. If the target is initially at low concentrations and/or if the number of probes is very large, and they have high affinity for the target, the DNA in solution may become depleted. In this paper we analyze the equilibrium and kinetics of hybridization of DNA biosensors in the case of strong target depletion, by extending the Langmuir adsorption model. We focus, in particular, on the detection of a small amount of a single-nucleotide "mutant" sequence (concentration c(2)) in a solution, which differs by one or more nucleotides from an abundant "wild-type" s equence (concentration c(1) >> c(2)). We show that depletion can give rise to a strongly enhanced sensitivity of the biosensors. Using representative values of rate constants and hybridization free energies, we find that in the depletion regime one could detect relative concentrations c(2)/c(1) that are up to 3 orders of magnitude smaller than in the conventional approach. The kinetics is surprisingly rich and exhibits a nonmonotonic adsorption with no counterpart in the no-depletion case. Finally, we show that, alongside enhanced detection sensitivity, this approach offers the possibility of sample enrichment, by substantially increasing the relative amount of the mutant over the wild-type sequence.