The Mu2e experiment at Fermilab will search for the neutrinoless μ−→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 5σ discovery sensitivity is Rμe=1.2×10−15, with a total expected background of 0.11±0.03 events. In the absence of a signal, the expected upper limit is Rμe<6.2×10−16 at 90% CL. This represents a three order of magnitude improvement over the current experimental limit of Rμe<7×10−13 at 90% CL set by the SINDRUM II experiment.