Abstract: In a special run of the LHC with beta star=2.5 km, proton-proton elastic-scattering events were recorded at root s=13 TeV with an integrated luminosity of 340 μb(-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 center dot 10(-4) GeV2 to -t=0.46 GeV2 using 6.9 million elastic-scattering candidates. This paper presents measurements of the total cross section sigma(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 -> 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(tot) and rho are sigma(tot) (PP -> X ) =104.7 +/- 1.1 mob , rho=0.098 +/- 0.011. The uncertainty in sigma(tot) is dominated by the luminosity measurement, and in rho by imperfect knowledge of the detector alignment and by modelling of the nuclear amplitude.