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In a special run of the LHC with $$\beta ^{\star } = 2.5$$ β ⋆ = 2.5  km, proton–proton elastic-scattering events were recorded at $$\sqrt{s} = 13$$ s = 13  TeV with an integrated luminosity of $$340~\upmu {\text {b}}^{-1}$$ 340 μ 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 $$-t = 2.5 \cdot 10^{-4}$$ - t = 2.5 · 10 - 4  GeV$$^{2}$$ 2 to $$-t = 0.46$$ - t = 0.46  GeV$$^{2}$$ 2 using 6.9 million elastic-scattering candidates. This paper presents measurements of the total cross section $$\sigma _{\text {tot}}$$ σ tot , parameters of the nuclear slope, and the $$\rho $$ ρ -parameter defined as the ratio of the real part to the imaginary part of the elastic-scattering amplitude in the limit $$t \rightarrow 0$$ t → 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 $$\sigma _{\text {tot}}$$ σ tot and $$\rho $$ ρ are $$\begin{aligned} \sigma _{\text {tot}}(pp\rightarrow X) = 104.7 \pm 1.1 \; \text{ mb },\quad \rho = 0.098 \pm 0.011 . \end{aligned}$$ σ tot ( p p → X ) = 104.7 ± 1.1 mb , ρ = 0.098 ± 0.011 . The uncertainty in $$\sigma _{\text {tot}}$$ σ tot is dominated by the luminosity measurement, and in $$\rho $$ ρ by imperfect knowledge of the detector alignment and by modelling of the nuclear amplitude.

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