Elastic and proton-dissociative ρ 0 photoproduction (γp → ρ 0 p, γp → ρ 0 N , respectively, with ρ 0 → π + π − ) has been studied in ep interactions at HERA for photon-proton centre-of-mass energies in the range 50 < W < 100 GeV and for |t| < 0.5 GeV 2 , where t is the square of the four-momentum transfer at the proton vertex; the results on the protondissociative reaction are presented for masses of the dissociated proton system in the range M 2 N < 0.1W 2 . For the elastic process, the π + π − invariant mass spectrum has been investigated as a function of t. As in fixed target experiments, the ρ 0 resonance shape is asymmetric; this asymmetry decreases with increasing |t|, as expected in models in which the asymmetry is ascribed to the interference of resonant and non-resonant π + π − production. The cross section has been studied as a function of W ; a fit to the resonant part with the form W a gives a = 0.16 ± 0.06 (stat.) +0.11 −0.15 (syst.). The resonant part of the γp → π + π − p cross section is 11.2 ± 0.1 (stat.) +1.1 −1.2 (syst.) µb at W = 71.7 GeV. The t dependence of the cross section can be described by a function of the type A ρ exp (−b ρ |t| + c ρ t 2 ) with b ρ = 10.9 ± 0.3 (stat.) +1.0 −0.5 (syst.) GeV −2 and c ρ = 2.7 ± 0.9 (stat.) +1.9 −1.7 (syst.) GeV −4 . The t dependence has also been studied as a function of W and a value of the slope of the pomeron trajectory α IP ′ = 0.23 ± 0.15 (stat.) +0.10 −0.07 (syst.) GeV −2 has been deduced. The ρ 0 spin density matrix elements r 04 00 , r 04 1−1 and ℜe[r 04 10 ] have been measured and found to be consistent with expectations based on s-channel helicity conservation. For proton-dissociative π + π − photoproduction in the ρ 0 mass range, the distributions of the two-pion invariant mass, W and the polar and azimuthal angles of the pions in the helicity frame are the same within errors as those for the elastic process. The t distribution has been fitted to an exponential function with a slope parameter 5.8 ± 0.3 (stat.) ± 0.5 (syst.) GeV −2 . The ratio of the elastic to proton-dissociative ρ 0 photoproduction cross section is 2.0 ± 0.2 (stat.) ± 0.7 (syst.).