  In a special run of the LHC with $\beta^\star = 2.5~$km, proton-proton
elastic-scattering events were recorded at $\sqrt{s} = 13~$TeV with an
integrated luminosity of $340~\mu \textrm{b}^{-1}$ using the ALFA subdetector
of ATLAS in 2016. The elastic cross section was measured differentially in the
Mandelstam $t$ variable in the range from $-t = 2.5 \cdot 10^{-4}~$GeV$^{2}$ to
$-t = 0.46~$GeV$^{2}$ using 6.9 million elastic-scattering candidates. This
paper presents measurements of the total cross section $\sigma_{\textrm{tot}}$,
parameters of the nuclear slope, and the $\rho$-parameter defined as the ratio
of the real part to the imaginary part of the elastic-scattering amplitude in
the limit $t \rightarrow 0$. These parameters are determined from a fit to the
differential elastic cross section using the optical theorem and different
parameterizations of the $t$-dependence. The results for
$\sigma_{\textrm{tot}}$ and $\rho$ are \begin{equation*}
\sigma_{\textrm{tot}}(pp\rightarrow X) = \mbox{104.7} \pm 1.1 \; \mbox{mb} , \;
\; \; \rho = \mbox{0.098} \pm 0.011 . \end{equation*} The uncertainty in
$\sigma_{\textrm{tot}}$ is dominated by the luminosity measurement, and in
$\rho$ by imperfect knowledge of the detector alignment and by modelling of the
nuclear amplitude.
