We present a compact, two-stage atomic beam source that produces a continuous, narrow, collimated and high-flux beam of rubidium atoms with sub-Doppler temperatures in three dimensions, which features very low emission of near-resonance fluorescence along the atomic trajectory. The atom beam source originates in a pushed two-dimensional magneto-optical trap (2D + MOT) feeding a slightly off-axis three-dimensional moving optical molasses stage that continuously cools and redirects the atom beam. The capture velocity of the moving optical molasses is deliberately chosen to be low, ∼ 3 m/s, to reduce fluorescence, and the cooling light is detuned by several atomic linewidths from resonance to reduce the absorption cross-section of cooling-induced fluorescence. Near-resonance light from the 2D + MOT and the push beam does not propagate to the output atomic trajectory due to a 10 • bend in the atomic trajectory. The atomic beam emitted from the two-stage source has a flux up to 1.6(3)×10 9 atoms/s, with an optimized temperature of 15.0(2) µK. We employ continuous Raman-Ramsey interference measurements at the atom beam output to study the sources of decoherence in the presence of continuous cooling, and demonstrate that the atom beam source effectively preserves high fringe contrast even during cooling. This cold-atom beam source is appropriate for use in atom interferometers and clocks, where continuous operation eliminates dead time, the slow atom beam velocity (6 -16 m/s) improves sensitivity, the narrow 3D velocity distribution improves fringe contrast, and the low reabsorption of scattered light mitigates decoherence caused by the continuous cooling process.