  One of the most important and well-established empirical results in astronomy
is the Kennicutt-Schmidt (KS) relation between the density of interstellar gas
and the rate at which that gas forms stars. A tight correlation between these
quantities has long been measured at galactic scales. More recently, using
surveys of YSOs, a KS relationship has been found within molecular clouds
relating the surface density of star formation to the surface density of gas;
however, the scaling of these laws varies significantly from cloud to cloud. In
this Letter, we use a recently developed, high-accuracy catalog of young
stellar objects from $\textit{Spitzer}$ combined with high-dynamic-range gas
column density maps of twelve nearby ($<$1.5 kpc) molecular clouds from
$\textit{Herschel}$ to re-examine the KS relation within individual molecular
clouds. We find a tight, linear correlation between clouds' star formation rate
per unit area and their gas surface density normalized by the gas free-fall
time. The measured intracloud KS relation, which relates star formation rate to
the volume density, extends over more than two orders of magnitude within each
cloud and is nearly identical in each of the twelve clouds, implying a constant
star formation efficiency per free-fall time $\epsilon_{\rm ff}\approx 0.026$.
The finding of a universal correlation within individual molecular clouds,
including clouds that contain no massive stars or massive stellar feedback,
favors models in which star formation is regulated by local processes such as
turbulence or stellar feedback such as protostellar outflows, and disfavors
models in which star formation is regulated only by galaxy properties or
supernova feedback on galactic scales.
