Elastic and proton-dissociative ρ0 photoproduction (γp → p0p, γp → ρ0N, respectively, with p0 → 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 GeV2, where t is the square of the four-momentum transfer at the proton vertex; the results on the proton-dissociative reaction are presented for masses of the dissociated proton system in the range M2N < 0.1W2. For the elastic process, the π+π- invariant mass spectrum has been investigated as a function of t. As in fixed target experiments, the ρ0 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 Wa 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 Aρ exp (-bρ|t| + cρt2) with bρ = 10.9 ± 0.3 (stat.) +1.0-0.5 (syst.) GeV-2 and cρ = 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 ρ0 spin density matrix elements r0400 r041-1 and ℜe[r0410] have been measured and found to be consistent with expectations based on s-channel helicity conservation. For proton-dissociative π+π- photoproduction in the ρ0 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 ρ0 photoproduction cross section is 2.0 ± 0.2 (stat.) ± 0.7 (syst.).