This dissertation describes a search for the decays B ⁰ → invisible and B⁰ → invisible+γ, where invisible refers to a final state consisting of long lived particles with a low cross-section for interaction with matter, leading to a low probability of detection in typical particle detectors. While the branching fractions for these decays predicted by the Standard Model are far below what could be feasably measured by current experiments, new physics such as right-handed neutrinos propagating in large extra space-time dimensions or light R-parity violating neutralinos in supersymmetry could greatly enhance the branching fractions. The decays are searched for in data corresponding to 423.5 fb ^-1 integrated luminosity produced at the ϒ(4S) resonance collected with the BABAR detector at the PEP-II B factory, corresponding to 2.30 × 10⁸ B⁰B¯⁰ pairs. Using those events that contain a hadronically reconstructed neutral B meson, evidence for the signal decays is sought in the remainder of the event. In (5.00 ± 0.02) × 10⁵ events with a fully reconstructed neutral B meson, a total of 39 events consistent with the B⁰ → invisible decay mode are seen in data with an expected background of 28.5 ± 7.8(stat.)±9.2(syst.) events, and 8 events consistent with the B⁰ → invisible+γ decay mode are seen in data mode with an expected background of 14.1 ± 5.5(stat.)±8.1(syst.) events, from which upper limits of [special characters omitted](B⁰ → invisible) < 11.7 × 10^-5 and of [special characters omitted](B⁰ → invisible+γ) < 4.3 × 10^-5 at the 90% confidence level are obtained.