  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.
