Photonic systems and technologies traditionally relegated to table-top experiments are poised to make the leap from the laboratory to real-world applications through integration. Stimulated Brillouin scattering (SBS) lasers, through their unique linewidth narrowing properties, are an ideal candidate to create highly-coherent waveguide integrated sources. In particular, cascaded-order Brillouin lasers show promise for multi-line emission, low-noise microwave generation and other optical comb applications. Photonic integration of these lasers can dramatically improve their stability to environmental and mechanical disturbances, simplify their packaging, and lower cost. While single-order silicon and cascade-order chalcogenide waveguide SBS lasers have been demonstrated, these lasers produce modest emission linewidths of 10-100 kHz. We report the first demonstration of a sub-Hz (~0.7 Hz) fundamental linewidth photonic-integrated Brillouin cascaded-order laser, representing a significant advancement in the state-of-the-art in integrated waveguide SBS lasers. This laser is comprised of a bus-ring resonator fabricated using an ultra-low loss Si3N4 waveguide platform. To achieve a sub-Hz linewidth, we leverage a high-Q, large mode volume, single polarization mode resonator that produces photon generated acoustic waves without phonon guiding. This approach greatly relaxes phase matching conditions between polarization modes, and optical and acoustic modes. Using a theory for cascaded-order Brillouin laser dynamics, we determine the fundamental emission linewidth of the first Stokes order by measuring the beat-note linewidth between and the relative powers of the first and third Stokes orders. Extension to the visible and near-IR wavebands is possible due to the low optical loss from 405 nm to 2350 nm, paving the way to photonic-integrated sub-Hz lasers for visible-light applications.