We carry out an independent search of Kepler photometry for small transiting planets with sizes 0.5-8.0 times that of Earth and orbital periods between 5 and 50 days, with the goal of measuring the fraction of stars harboring such planets. We use a new transit search algorithm, TERRA, optimized to detect small planets around photometrically quiet stars. We restrict our stellar sample to include the 12000 stars having the lowest photometric noise in the Kepler survey, thereby maximizing the detectability of Earth-size planets. We report 129 planet candidates having radii less than 6R_E_ found in three years of Kepler photometry (quarters 1-12). Forty-seven of these candidates are not in Batalha et al. (J/ApJS/204/24), which only analyzed photometry from quarters 1-6. We gather Keck HIRES spectra for the majority of these targets leading to precise stellar radii and hence precise planet radii. We make a detailed measurement of the completeness of our planet search. We inject synthetic dimmings from mock transiting planets into the actual Kepler photometry. We then analyze that injected photometry with our TERRA pipeline to assess our detection completeness for planets of different sizes and orbital periods. We compute the occurrence of planets as a function of planet radius and period, correcting for the detection completeness as well as the geometric probability of transit, R_*_/a. The resulting distribution of planet sizes exhibits a power law rise in occurrence from 5.7R_E_ down to 2R_E_, as found in Howard et al. (2012ApJS..201...15H). That rise clearly ends at 2R_E_. The occurrence of planets is consistent with constant from 2R_E_ toward 1R_E_. This unexpected plateau in planet occurrence at 2R_E_ suggests distinct planet formation processes for planets above and below 2R_E_.