  The discovery of quasars at $z \gtrsim$ 7 poses serious challenges because it
is not known how 10$^9$ \Ms\ black holes formed by this epoch. The leading
candidates for the seeds of these quasars are 10$^5$ \Ms\ direct-collapse black
holes forming in atomically cooled haloes at $z \sim$ 15 - 20. However, the
Lyman-Werner (LW) UV backgrounds required to form such objects are extreme,
$\gtrsim$ 10$^4$ $\rm J_{21}$, and may have been rare in the early universe.
Here, we model the formation of Population III stars in moderate LW backgrounds
of 100 and 500 $\rm J_{21}$ that were much more common at early times. We find
that these backgrounds allow haloes 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$_2$. At the onset of collapse, Ly$\alpha$ cooling dominates
in the outer regions of the halo but H$_2$ cooling regulates the collapse of
the core, at rates that are 10 - 50 times those in minihaloes because of higher
virial temperatures. Supercharged H$_2$ cooling leads to the formation of 1800
- 2800 \Ms\ primordial stars, with radiative feedback from the star halting
accretion and setting its upper limit in mass. Such stars may lead to a
population of less-massive, lower luminosity quasars that could be discovered
by the {\em James Webb Space Telescope}, {\em Euclid} and the {\em Roman Space
Telescope} in the coming decade.
