© 2019. The American Astronomical Society. All rights reserved.. We present the analysis underpinning the measurement of cosmological parameters from 207 spectroscopically classified SNe Ia from the first 3 years of the Dark Energy Survey Supernova Program (DES-SN), spanning a redshift range of 0.017 < z < 0.849. We combine the DES-SN sample with an external sample of 122 low-redshift (z < 0.1) SNe Ia, resulting in a "DES-SN3YR" sample of 329 SNe Ia. Our cosmological analyses are blinded: after combining our DES-SN3YR distances with constraints from the Cosmic Microwave Background, our uncertainties in the measurement of the dark energy equation-of-state parameter, w, are 0.042 (stat) and 0.059 (stat+syst) at 68% confidence. We provide a detailed systematic uncertainty budget, which has nearly equal contributions from photometric calibration, astrophysical bias corrections, and instrumental bias corrections. We also include several new sources of systematic uncertainty. While our sample is less than one-third the size of the Pantheon sample, our constraints on w are only larger by 1.4×, showing the impact of the DES-SN Ia light-curve quality. We find that the traditional stretch and color standardization parameters of the DES-SNe Ia are in agreement with earlier SN Ia samples such as Pan-STARRS1 and the Supernova Legacy Survey. However, we find smaller intrinsic scatter about the Hubble diagram (0.077 mag). Interestingly, we find no evidence for a Hubble residual step (0.007 ± 0.018 mag) as a function of host-galaxy mass for the DES subset, in 2.4σ tension with previous measurements. We also present novel validation methods of our sample using simulated SNe Ia inserted in DECam images and using large catalog-level simulations to test for biases in our analysis pipelines.