The Cryogenic Dark Matter Search low ionization threshold experiment (CDMSlite) searches for interactions between dark matter particles and germanium nuclei in cryogenic detectors. The experiment has achieved a low energy threshold with improved sensitivity to low-mass (<10 GeV/c$^2$) dark matter particles. We present an analysis of the final CDMSlite data set, taken with a different detector than was used for the two previous CDMSlite data sets. This analysis includes a data salting method to protect against bias, improved noise discrimination, background modeling, and the use of profile likelihood methods to search for a dark matter signal in the presence of backgrounds. We achieve an energy threshold of 70 eV and significantly improve the sensitivity for dark matter particles with masses between 2.5 and 10 GeV/c$^2$ compared to previous analyses. We set an upper limit on the dark matter-nucleon scattering cross section in germanium of 5.4$times$10$^{-42}$ cm$^2$ at 5 GeV/c$^2$, a factor of $sim$2.5 improvement over the previous CDMSlite result.
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