  We present the results of N$_2$H$^+$ ($J=1-0$) observations toward Serpens
South, the nearest cluster-forming, infrared dark cloud. The physical
quantities are derived by fitting the hyperfine structure of N$_2$H$^+$. The
Herschel and 1.1-mm continuum maps show that a pc-scale filament fragments into
three clumps with radii of $0.1-0.2$ pc and masses of $40-230M_\odot$. We find
that the clumps contain smaller-scale ($\sim 0.04$ pc) structures, i.e., dense
cores. We identify 70 cores by applying CLUMPFIND to the N$_2$H$^+$ data cube.
In the central cluster-forming clump, the excitation temperature and line-width
tend to be large, presumably due to protostellar outflow feedback and stellar
radiation. However, for all the clumps, the virial ratios are evaluated to be
$0.1-0.3$, indicating that the internal motions play only a minor role in the
clump support. The clumps exhibit no free-fall, but low-velocity infall, and
thus the clumps should be supported by additional forces. The most promising
force is the globally-ordered magnetic field observed toward this region. We
propose that the Serpens South filament was close to magnetically-critical and
ambipolar diffusion triggered the cluster formation. We find that the northern
clump, which shows no active star formation, has a mass and radius comparable
to the central cluster-forming clump, and therefore, it is a likely candidate
of a {\it pre-protocluster clump}. The initial condition for cluster formation
is likely to be a magnetically-supported clump of cold, quiescent gas. This
appears to contradict the accretion-driven turbulence scenario, for which the
turbulence in the clumps is maintained by the accretion flow.
