  We explore possibilities of collapse and star formation in Population III
objects exposed to the external ultraviolet background (UVB) radiation.
Assuming spherical symmetry, we solve self-consistently radiative transfer of
photons, non-equilibrium H2 chemistry, and gas hydrodynamics. Although the UVB
does suppress the formation of low mass objects, the negative feedback turns
out to be weaker than previously suggested. In particular, the cut-off scale of
collapse drops significantly below the virial temperature 10^4 K at weak UV
intensities, due to both self-shielding of the gas and H2 cooling. Clouds above
this cut-off tend to contract highly dynamically, further promoting
self-shielding and H2 formation. For plausible radiation intensities and
spectra, the collapsing gas can cool efficiently to temperatures well below
10^4 K before rotationally supported and the final H2 fraction reaches 10^{-3}.
Our results imply that star formation can take place in low mass objects
collapsing in the UVB. The threshold baryon mass for star formation is \sim
10^9 solar mass for clouds collapsing at redshifts z \simlt 3, but drops
significantly at higher redshifts. In a conventional cold dark matter universe,
the latter coincides roughly with that of the 1 \sigma density fluctuations.
Objects near and above this threshold can thus constitute `building blocks' of
luminous structures, and we discuss their links to dwarf spheroidal/elliptical
galaxies and faint blue objects. These results suggest that the UVB can play a
key role in regulating the star formation history of the Universe.
