We present the first cosmological parameter constraints using measurements of type Ia supernovae (SNe Ia) from the Dark Energy Survey Supernova Program (DES-SN). The analysis uses a subsample of 207 spectroscopically confirmed SNe Ia from the first three years of DES-SN, combined with a low-redshift sample of 122 SNe from the literature. Our "DES-SN3YR" result from these 329 SNe Ia is based on a series of companion analyses and improvements covering SN Ia discovery, spectroscopic selection, photometry, calibration, distance bias corrections, and evaluation of systematic uncertainties. For a flat ΛCDM model we find a matter density Ω m = 0.331 ± 0.038. For a flat wCDM model, and combining our SN Ia constraints with those from the cosmic microwave background (CMB), we find a dark energy equation of state w = -0.978 ± 0.059, and ω m = 0.321 ± 0.018. For a flat w 0 w a CDM model, and combining probes from SN Ia, CMB and baryon acoustic oscillations, we find w0 = -0.885 ± 0.114 and w a = -0.387 ± 0.430. These results are in agreement with a cosmological constant and with previous constraints using SNe Ia (Pantheon, JLA).