The rate at which massive stars eject mass in stellar winds significantly influences their evolutionary path. Cosmic rates of nucleosynthesis, explosive stellar phenomena, and compact object genesis depend on this poorly known facet of stellar evolution. We employ an unexploited observational technique for measuring the mass-loss rates of O- and early-B stars. Our approach, which has no adjustable parameters, uses the principle of pressure equilibrium between the stellar wind and the ambient interstellar medium for a high-velocity star generating an infrared bowshock nebula. Results for twenty bowshock-generating stars show good agreement with two sets of theoretical predictions for O5--O9.5 main-sequence stars, yielding $dot M=$1.3$times$10$^{-6}$ to 2$times$10$^{-9}$ solar masses per year. Although $dot M$ values derived for this sample are smaller than theoretical expectations by a factor of about two, this discrepancy is greatly reduced compared to canonical mass-loss methods. Bowshock-derived mass-loss rates are factors of ten smaller than H$alpha$-based measurements (uncorrected for clumping) for similar stellar types and are nearly an order of magnitude larger than P$^{4+}$ and some other UV absorption-line-based diagnostics. Ambient interstellar densities of at least several cm$^{-3}$ appear to be required for formation of a prominent infrared bowshock nebula. $dot M$ measurements for early-B stars are not yet compelling owing to the small number in our sample and the lack of clear theoretical predictions in the regime of lower stellar luminosities. These results may constitute a partial resolution of the extant weak-wind problem for late-O stars. The technique shows promise for determining mass-loss rates in the weak-wind regime.
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