  The Mu2e experiment at Fermilab will search for the neutrinoless $\mu^- \to
e^-$ conversion in the field of an aluminum nucleus. The Mu2e data-taking plan
assumes two running periods, Run I and Run II, separated by an approximately
two-year-long shutdown. This paper presents an estimate of the expected Mu2e
Run I search sensitivity and includes a detailed discussion of the background
sources, uncertainties of their prediction, analysis procedures, and the
optimization of the experimental sensitivity. The expected Run I $5 \sigma$
discovery sensitivity is $R_{\mu e} = 1.2 \times 10^{-15}$, with a total
expected background of $0.11 \pm 0.03$ events. In the absence of a signal, the
expected upper limit is $R_{\mu e} < 6.2 \times 10^{-16}$ at 90% CL. This
represents a three order of magnitude improvement over the current experimental
limit of $R_{\mu e} < 7 \times 10^{-13}$ at 90\% CL set by the SINDRUM II
experiment.
