23 pages, 4 figures, submitted to Phys. Rev. D (Rapid Communication) ; We report the first measurement of the top quark mass using the decay length technique in ppbar collisions at a center-of-mass energy of 1.96 TeV. This technique uses the measured flight distance of the b hadron to infer the mass of the top quark in lepton plus jets events with missing transverse energy. It relies solely on tracking and avoids the jet energy scale uncertainty that is common to all other methods used so far. We apply our novel method to a 695 pb^-1 data sample recorded by the CDF II detector at Fermilab and extract a measurement of m_t = 180.7^{+15.5}_{-13.4} (stat.) ± 8.6 (syst.) GeV/c^2. While the uncertainty of this result is larger than that of other measurements, the dominant uncertainties in the decay length technique are uncorrelated with those in other methods. This result can help reduce the overall uncertainty when combined with other existing measurements of the top quark mass.