We investigate the contribution of star-forming galaxies to the ionizing background at z \sim 3 , building on previous work based on narrowband ( NB3640 ) imaging in the SSA22a field . We use new Keck/LRIS spectra of Lyman break galaxies ( LBGs ) and narrowband-selected Ly \alpha emitters ( LAEs ) to measure redshifts for 16 LBGs and 87 LAEs at z > 3.055 , such that our NB3640 imaging probes the Lyman-continuum ( LyC ) region . When we include the existing set of spectroscopically-confirmed LBGs , our total sample with z > 3.055 consists of 41 LBGs and 91 LAEs , of which nine LBGs and 20 LAEs are detected in our NB3640 image . With our combined imaging and spectroscopic data sets , we critically investigate the origin of NB3640 emission for detected LBGs and LAEs . We remove from our samples 3 LBGs and 3 LAEs with spectroscopic evidence of contamination of their NB3640 flux by foreground galaxies , and statistically model the effects of additional , unidentified foreground contaminants . The resulting contamination and LyC-detection rates , respectively , are 62 \pm 13 % and 8 \pm 3 % for our LBG sample , and 47 \pm 10 % and 12 \pm 2 % for our LAE sample . The corresponding ratios of non-ionizing UV to LyC flux-density , corrected for intergalactic medium ( IGM ) attenuation , are 18.0 ^ { +34.8 } _ { -7.4 } for LBGs , and 3.7 ^ { +2.5 } _ { -1.1 } for LAEs . We use these ratios to estimate the total contribution of star-forming galaxies to the ionizing background and the hydrogen photoionization rate in the IGM , finding values larger than , but consistent with , those measured in the Ly \alpha forest . Finally , the measured UV to LyC flux-density ratios imply model-dependent LyC escape fractions of f _ { esc } ^ { LyC } \sim 5 - 7 % for our LBG sample and f _ { esc } ^ { LyC } \sim 10 - 30 % for our fainter LAE sample .