We present the results of a search for outer Solar System objects in the full six years of data (Y6) from the Dark Energy Survey (DES). The DES covered a contiguous $5000$ deg$^2$ of the southern sky with $approx 80,000$ $3$ deg$^2$ exposures in the $grizY$ optical/IR filters between 2013 and 2019. This search yielded 815 trans-Neptunian objects (TNOs), one Centaur and one Oort cloud comet, with 461 objects reported for the first time in this paper. We present methodology that builds upon our previous search carried out on the first four years of data. Here, all DES images were reprocessed with an improved detection pipeline that leads to an average completeness gain of 0.47 mag per exposure, as well as an improved transient catalog production and optimized algorithms for linkage of detections into orbits. All objects were verified by visual inspection and by computing the sub-threshold significance, the total signal-to-noise ratio in the stack of images in which the objects presence is indicated by the orbit fit, but no detection was reported. This yields a highly pure catalog of TNOs complete to $r approx 23.8$ mag and distances $29<d<2500$ au. The Y6 TNOs have minimum (median) of 7 (12) distinct nights detections and arcs of 1.1 (4.2) years, and will have $grizY$ magnitudes available in a further publication. We present software for simulating our observational biases that enable comparisons of population models to our detections. Initial inferences demonstrating the statistical power of the DES catalog are: the data are inconsistent with the CFEPS-L7 model for the classical Kuiper Belt; the 16 ``extreme TNOs ($a>150$ au, $q>30$ au) are consistent with the null hypothesis of azimuthal isotropy; and non-resonant TNOs with $q>38$ au, $a>50$ au show a highly significant tendency to be sunward of the major mean motion resonances, whereas this tendency is not present for $q<38$ au.
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