  As part of the T-REX project, a number of R&D and prototyping activities have
been carried out during the last years to explore the applicability of gaseous
Micromegas-read TPCs in rare event searches like double beta decay (DBD), axion
research and low-mass WIMP searches. In both this and its companion paper, we
compile the main results of the project and give an outlook of application
prospects for this detection technique. While in the companion paper we focus
on axions and WIMPs, in this paper we focus on the results regarding the
measurement of the DBD of $^{136}$Xe in a high pressure Xe (HPXe) TPC.
Particularly relevant are the results obtained in Xe + TMA mixtures with
microbulk Micromegas, showing very promising results in terms of gain,
stability of operation, and energy resolution at pressures up to 10 bar. TMA at
levels of $\sim$1\% reduces electron diffusion by a factor of 10 with respect
to pure Xe, improving the quality of the topological pattern, with a positive
impact on the discrimination capability. Operation with a medium size prototype
(30 cm diameter and 38 cm drift) of 1 kg of Xe at 10 bar in the fiducial
volumen has allowed to test the detection concept in realistic experimental
conditions. Microbulk Micromegas are able to image the DBD ionization signature
with high quality while, at the same time, measuring its energy deposition with
a resolution of at least $\sim$3% FWHM @ Q$_{\beta\beta}$, and probably down to
$\sim$1% FWHM. In addition, first results on the topological signature
information show promising background discrimination capabilities out of reach
of other experimental implementations. Moreover, microbulk Micromegas have very
low levels of intrinsic radioactivity, and offer cost-effective scaling-up
options. All these results demonstrate that Micromegas-read HPXe TPC is a very
competitive technique for the next generation DBD experiments.
