  We present a new analysis of the minimum mass for star formation, based on
opacity-limited fragmentation. Our analysis differs from the standard one,
which considers hierarchical fragmentation of a 3-D medium, and yields
$M_{_{\rm MIN}} \sim 0.007 {\rm to} 0.010 M_\odot$ for Population I star
formation. Instead we analyse the more realistic situation in which there is
one-shot fragmentation of a shock-compressed layer, of the sort which arises in
turbulent star-forming clouds. In this situation, $M_{_{\rm MIN}}$ can be
smaller than $0.003 M_\odot$. Our analysis is more stringent than the standard
one in that (a) it requires fragments to have condensation timescales shorter
than all competing mass scales, and (b) it takes into acount that a fragment
grows by accretion whilst it is condensing out, and has to radiate away the
energy dissipated in the associated accretion shock. It also accords with the
recent detection, in young star clusters, of free-floating star-like objects
having masses as low as $0.003 {\rm M}_\odot$.
