This thesis describes a search for the double beta (ββ)decay of^(76) Ge using an array of eight high purity, high resolution germanium detectors containing a total of 3.9 x 10^(24 76)Ge nuclei. There are three modes of (ββ) decay: neutrinoless (Oѵ) (ββ) decay, (ββ) decay with Majoron emission (x^0 (ββ) decay) and two neutrino (2ѵ) (ββ) decay. The first two modes violate lepton number conservation, while the third is allowed by the Standard Model. Oѵ (ββ)decay may take place to the ground state of the daughter nucleus (0^+ → 0^+ transition) or to an excited state of the daughter nucleus (0^+ → 2^+ transition). The detector was operated for a total time of 2033 h (0.23 y) which translates to 1.30 kg-y. The background at the ββ decay transition energy was 0.53 counts keV^(-1) y^(-1) (10^(23 76)Ge nuclei)^(-1) . No evidence for double beta decay of any sort was found and the half life limits are T^(oѵ)/(1/2) (0^+ → 0^+) > 1.2 x 10^(23) y, T^(oѵ)/(1/2) (0^+ → 2^+) > 2 x 10^(22) y, T^(x^o)/(1/2) > 8 x 10^(20), and T^(2ѵ)/(1/2) > 2 x 10^(20)y, all at 90% c.l. The limit for the neutrinoless mode translates to an upper limit of between 16 and 1.6 eV on the Majorana mass of the neutrino, depending on the nuclear matrix element used.