We present legacy cosmic shear measurements and cosmological constraints using six years of Dark Energy Survey (DES) imaging data. From these data, we study $$140milliongalaxies(8.29galaxies/arcmin$^2$) that are 50% complete at $i=24.0$ mag and extend beyond $z=1.2$. We divide the galaxies into four redshift bins, and obtain cosmic shear measurement with a signal-to-noise of 83, a factor of 2 higher than the Year 3 analysis. We model the uncertainties due to shear and redshift calibrations, and discard measurements on small angular scales to mitigate baryon feedback and other small-scale uncertainties. We consider two fiducial models to account for the intrinsic alignment of the galaxies. We conduct a blind analysis in the context of the ~model and find $S_8 \equiv \sigma_8(\Omega_{\rm m}/0.3)^{0.5} = 0.798^{+0.014}_{-0.015}$ (marginalized mean with 68$\%$ confidence limits) when using the non-linear alignment model (NLA) and $S_8={0.783}_{-0.015}^{+0.019}$ with the tidal alignment and tidal torque model (TATT), providing 1.8% and 2.5% uncertainty on $S_8$. Compared to constraints from the cosmic microwave background from {} 2018, ACT DR6 and SPT-3G DR1, we find consistency in the full parameter space at 1.1$\sigma$ (1.7$\sigma$) and in $S_8$ at 2.0$\sigma$ (2.3$\sigma$) for NLA (TATT), respectively. The result using the NLA model is preferred according to the Bayesian evidence. We find that the model choice for intrinsic alignment and baryon feedback can impact the value of our $S_8$ constraint up to $1\sigma$. For our fiducial model choices, the resultant uncertainties in $S_8$ are primarily degraded by the removal of scales, as well as the marginalization over the intrinsic alignment parameters. We demonstrate that our result is internally consistent and robust to different choices in calibrating the data, owing to methodological improvements in shear and redshift measurement, which lay the foundation for next-generation cosmic shear programs.