  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 object's 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.
