The proton–proton elastic differential cross section at very small four momentum transfer squared has been measured at three different incident proton momenta in the range of 2.5 to 3.2 GeV/c by detecting the recoil proton at polar angles close to ${90}^{\circ }$. The measurement was performed at COSY with the KOALA detector covering the Coulomb–nuclear interference region. The total cross section ${\sigma }_{\text{tot}}$, which has been determined precisely, is consistent with previous measurements. The values of the slope parameter B and the relative real amplitude ratio ρ determined in this experiment alleviate the lack of data in the relevant energy region. These precise data on ρ might be an important check for a new dispersion analysis.