  We present the most precise estimate to date of the clustering of quasars on
very small scales, based on a sample of 47 binary quasars with magnitudes of
$g<20.85$ and proper transverse separations of $\sim 25\,h^{-1}$\,kpc. Our
sample of binary quasars, which is about 6 times larger than any previous
spectroscopically confirmed sample on these scales, is targeted using a Kernel
Density Estimation technique (KDE) applied to Sloan Digital Sky Survey (SDSS)
imaging over most of the SDSS area. Our sample is "complete" in that all of the
KDE target pairs with $17.0 \lesssim R \lesssim 36.2\,h^{-1}$\,kpc in our area
of interest have been spectroscopically confirmed from a combination of
previous surveys and our own long-slit observational campaign. We catalogue 230
candidate quasar pairs with angular separations of $<8\arcsec$, from which our
binary quasars were identified. We determine the projected correlation function
of quasars ($\bar W_{\rm p}$) in four bins of proper transverse scale over the
range $17.0 \lesssim R \lesssim 36.2\,h^{-1}$\,kpc. The implied small-scale
quasar clustering amplitude from the projected correlation function, integrated
across our entire redshift range, is $A=24.1\pm3.6$ at $\sim 26.6
~h^{-1}$\,kpc. Our sample is the first spectroscopically confirmed sample of
quasar pairs that is sufficiently large to study how quasar clustering evolves
with redshift at $\sim 25 ~h^{-1}$ kpc. We find that empirical descriptions of
how quasar clustering evolves with redshift at $\sim 25 ~h^{-1}$ Mpc also
adequately describe the evolution of quasar clustering at $\sim 25 ~h^{-1}$
kpc.
