  The leading contenders for the seeds of $z > 6$ quasars are direct-collapse
black holes (DCBHs) forming in atomically-cooled halos at $z \sim$ 20. However,
the Lyman-Werner (LW) UV background required to form DCBHs of 10$^5$ \Ms\ are
extreme, about 10$^4$ J$_{21}$, and may have been rare in the early universe.
Here we investigate the formation of intermediate-mass black holes (IMBHs)
under moderate LW backgrounds of 100 and 500 J$_{21}$ that were much more
common at early times. These backgrounds allow halos to grow to a few 10$^6$ -
10$^7$ \Ms\ and virial temperatures of nearly 10$^4$ K before collapsing but do
not completely sterilize them of H$_2$. Gas collapse then proceeds via
Ly$\alpha$ and rapid H$_2$ cooling at rates that are 10 - 50 times those in
normal Pop III star-forming haloes but less than those in purely
atomically-cooled haloes. Pop III stars accreting at such rates become blue and
hot, and we find that their ionizing UV radiation limits their final masses to
1800 - 2800 \Ms\, at which they later collapse to IMBHs. Moderate LW
backgrounds thus produced IMBHs in far greater numbers than DCBHs in the early
universe.
