  We present the angular diameter distance measurement obtained with the
Baryonic Acoustic Oscillation feature from galaxy clustering in the completed
Dark Energy Survey, consisting of six years (Y6) of observations. We use the Y6
BAO galaxy sample, optimized for BAO science in the redshift range 0.6<$z$<1.2,
with an effective redshift at $z_{\rm eff}$=0.85 and split into six tomographic
bins. The sample has nearly 16 million galaxies over 4,273 square degrees. Our
consensus measurement constrains the ratio of the angular distance to sound
horizon scale to $D_M(z_{\rm eff})/r_d$ = 19.51$\pm$0.41 (at 68.3% confidence
interval), resulting from comparing the BAO position in our data to that
predicted by Planck $\Lambda$CDM via the BAO shift parameter
$\alpha=(D_M/r_d)/(D_M/r_d)_{\rm Planck}$. To achieve this, the BAO shift is
measured with three different methods, Angular Correlation Function (ACF),
Angular Power Spectrum (APS), and Projected Correlation Function (PCF)
obtaining $\alpha=$ 0.952$\pm$0.023, 0.962$\pm$0.022, and 0.955$\pm$0.020,
respectively, which we combine to $\alpha=$ 0.957$\pm$0.020, including
systematic errors. When compared with the $\Lambda$CDM model that best fits
Planck data, this measurement is found to be 4.3% and 2.1$\sigma$ below the
angular BAO scale predicted. To date, it represents the most precise angular
BAO measurement at $z$>0.75 from any survey and the most precise measurement at
any redshift from photometric surveys. The analysis was performed blinded to
the BAO position and it is shown to be robust against analysis choices, data
removal, redshift calibrations and observational systematics.
