  A continuous guided atomic beam of $^{88}\mathrm{Sr}$ with a phase-space
density exceeding $10^{-4}$ in the moving frame and a flux of $3 \times 10^{7}
\, \mathrm{at \, s^{-1}}$ is demonstrated. This phase-space density is around
three orders of magnitude higher than previously reported for steady-state
atomic beams. We detail the architecture necessary to produce this ultracold
atom source and characterize its output after $\sim 4 \, \mathrm{cm}$ of
propagation. With radial temperatures of less than $1 \,\mu \mathrm{K}$ and a
velocity of $8.4 \, \mathrm{cm \, s^{-1}}$ this source is ideal for a range of
applications. For example, it could be used to replenish the gain medium of an
active optical superradiant clock or be employed to overcome the Dick effect
that can limit the performance of pulsed-mode atom interferometers, atomic
clocks and ultracold atom based sensors in general. Finally, this result
represents a significant step towards the development of a steady-state atom
laser.
