  We study the prospects for observing H$_2$ emission during the assembly of
primordial molecular cloud kernels. The primordial molecular cloud cores, which
resemble those at the present epoch, can emerge around $1+z=20$ according to
recent numerical simulations. The kernels form inside the cores, and the first
stars will appear inside the kernels. A kernel typically contracts to form one
of the first generation stars with an accretion rate that is as large as $\sim
0.01 M_\odot$ year$^{-1}$. This occurs due to the primordial abundances that
result in a kernel temperature of order 1000K, and the collapsing kernel emits
H$_2$ line radiation at a rate $\sim 10^{35}$ erg sec$^{-1}$. Principally
$J=5-3$ (v=0) rotational emission of H$_2$ is expected. At redshift $1+z=20$,
the expected flux is $\sim 0.01~\mu$Jy for a single kernel. While an individual
object is not observable by any facilities available in the near future, the
expected assembly of primordial star clusters on sub-galactic scales can result
in fluxes at the sub-mJy level. This is marginally observable with ASTRO-F. We
also examine the rotational $J=2-0$ (v=0) and vibrational $\delta v = 1$
emission lines. The former may possibly be detectable with ALMA.
