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 arcseccomplete arcsec 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 $<8arcsec$, 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 $A=24.1pm3.6$ 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.