A measurement of the t-channel single-top-quark and single-top-antiquark production cross-sections in the lepton+jets channel is presented, using 3.2 fb−1 of proton-proton collision data at a centre-of-mass energy of 13 TeV, recorded with the ATLAS detector at the LHC in 2015. Events are selected by requiring one charged lepton (electron or muon), missing transverse momentum, and two jets with high transverse momentum, exactly one of which is required to be b-tagged. Using a binned maximum-likelihood fit to the discriminant distribution of a neural network, the cross-sections are determined to be σ(tq) = 156 ± 5 (stat.) ± 27 (syst.) ± 3 (lumi.) pb for single top-quark production and σ(t¯ q) = 91 ± 4 (stat.) ± 18 (syst.) ± 2 (lumi.) pb for single top-antiquark production, assuming a top-quark mass of 172.5 GeV. The cross-section ratio is measured to be Rt=σ(tq)/σ(t¯q)=1.72±0.09 (stat.) ± 0.18 (syst.). All results are in agreement with Standard Model predictions.[Figure not available: see fulltext.]