We measured the analyzing power A out to ${\mathit{P}}_{\mathrm{\ensuremath{\perp}}}^{2}$=7.1 (GeV/c${)}^{2}$ with high precision by scattering a 24-GeV/c unpolarized proton beam from the new University of Michigan polarized proton target; the target's 1-W cooling power allowed a beam intensity of more than 2\ifmmode\times\else\texttimes\fi{}${10}^{11}$ protons per pulse. This high beam intensity together with the unexpectedly high average target polarization of about 85% allowed unusually accurate measurements of A at large ${\mathit{P}}_{\mathrm{\ensuremath{\perp}}}^{2}$. These precise data confirmed that the one-spin parameter A is nonzero and indeed quite large at high ${\mathit{P}}_{\mathrm{\ensuremath{\perp}}}^{2}$; most theoretical models predict that A should go to zero.