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 $340~\upmu {\text {b}}^{-1}$ 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 $\sigma _{\text {tot}}$, 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 $\sigma _{\text {tot}}$ and $\rho $ are $\begin{aligned} \sigma _{\text {tot}}(pp\rightarrow X) = 104.7 \pm 1.1 \; \text{ mb },\quad \rho = 0.098 \pm 0.011 . \end{aligned}$The uncertainty in $\sigma _{\text {tot}}$ is dominated by the luminosity measurement, and in $\rho $ by imperfect knowledge of the detector alignment and by modelling of the nuclear amplitude.