The first data taken with the final setup of the HEIDELBERG-MOSCOW double beta decay experiment are presented. This experiment gives at present the most stringent upper bound, improving the neutrino mass limit into the sub-eV range. Out of 19.2 kg of 86% enriched Ge-76 five crystals were grown with a total mass of 11.51 kg. Since February 1995 all five detectors, corresponding to 10.96 kg active mass, are in regular operation in a common shielding in the Gran Sasso underground laboratory. Since no signal is seen for the neutrinoless double beta decay (0 nu beta<beta), the measured data with a statistical significance of 12.50 kg.y result in a new half life Limit of T-1/2(0(+) --> 0(+)) > 6.4.10(24)y. With this limit a Majorana mass of the neutrinos larger than 0.6 eV (90% C.L.) is excluded. From the data taken in the previously operated setup with three enriched detectors in a common shielding and a statistical significance of 10.58 kg.y new results are extracted for the two neutrino double beta decay (2 nu beta beta) of Ge-76. The procedure of a quantitative and model-independant description of the background is described in some detail. The result is T-1/2(2 nu) = (1.77(-0.01)(+0.01)(stat)(-0.11)(+0.13)(syst)).10(21)y. Furtheron new half life limits of various Majoron models are deduced. Future improvements on the background with the application of Digital Pulse Shape Analysis are discussed.