  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, $\sim 3$ 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 degree
bend in the atomic trajectory. The atomic beam emitted from the two-stage
source has a flux up to $1.6(3)\times 10^9\;\textrm{atoms/s}$, with an
optimized temperature of $15.0(2)\;\mu$K. 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.
