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 $rho$ determined in this experiment alleviate the lack of data in the relevant energy region. This precise data on $rho$ might be an important check for a new dispersion analysis.