This work, together with its companion paper, Secco, Samuroff et al. [Phys. Rev. D 105, 023515 (2022)], present the Dark Energy Survey Year 3 cosmic-shear measurements and cosmological constraints based on an analysis of over 100 million source galaxies. With the data spanning 4143 deg(2) on the sky, divided into four redshift bins, we produce a measurement with a signal-to-noise of 40. We conduct a blind analysis in the context of the Lambda-Cold Dark Matter (Lambda CDM) model and find a 3% constraint of the clustering amplitude, S-8 sigma(8)(Omega(m)/0.3)(0.5) = 0.759(-0.023)(+0.025). A Lambda CDM-Optimized analysis, which safely includes smaller scale information, yields a 2% precision measurement of S-8 = 0.772(-0.017)(+0.018) that is consistent with the fiducial case. The two low-redshift measurements are statistically consistent with the Planck Cosmic Microwave Background result, however, both recovered S-8 values are lower than the high-redshift prediction by 2.3 sigma and 2.1 sigma (p-values of 0.02 and 0.05), respectively. The measurements are shown to be internally consistent across redshift bins, angular scales and correlation functions. The analysis is demonstrated to be robust to calibration systematics, with the S-8 posterior consistent when varying the choice of redshift calibration sample, the modeling of redshift uncertainty and methodology. Similarly, we find that the corrections included to account for the blending of galaxies shifts our best-fit S-8 by 0.5 sigma without incurring a substantial increase in uncertainty. We examine the limiting factors for the precision of the cosmological constraints and find observational systematics to be subdominant to the modeling of astrophysics. Specifically, we identify the uncertainties in modeling baryonic effects and intrinsic alignments as the limiting systematics.