Exclusive photoproduction of $rho^0(770)$ mesons is studied using the H1 detector at the $ep$ collider HERA. A sample of about 900000 events is used to measure single- and double-differential cross sections for the reaction $gamma p to pi^{+}pi^{-}Y$. Reactions where the proton stays intact (${m_Y{=}m_p}$) are statistically separated from those where the proton dissociates to a low-mass hadronic system ($m_p{<}m_Y{<}10$ GeV). The double-differential cross sections are measured as a function of the invariant mass $m_{pipi}$ of the decay pions and the squared $4$-momentum transfer $t$ at the proton vertex. The measurements are presented in various bins of the photon-proton collision energy $W_{gamma p}$. The phase space restrictions are $0.5 < m_{pipi} < 2.2$ GeV, ${vert tvert < 1.5}$ GeV${}^2$, and ${20 < W_{gamma p} < 80}$ GeV. Cross section measurements are presented for both elastic and proton-dissociative scattering. The observed cross section dependencies are described by analytic functions. Parametrising the $m_{pipi}$ dependence with resonant and non-resonant contributions added at the amplitude level leads to a measurement of the $rho^{0}(770)$ meson mass and width at $m_rho = 770.8 {}^{+2.6}_{-2.7}$ (tot) MeV and $Gamma_rho = 151.3 {}^{+2.7}_{-3.6}$ (tot) MeV, respectively. The model is used to extract the $rho^0(770)$ contribution to the $pi^{+}pi^{-}$ cross sections and measure it as a function of $t$ and $W_{gamma p}$. In a Regge asymptotic limit in which one Regge trajectory $alpha(t)$ dominates, the intercept $alpha(t{=}0) = 1.0654 {}^{+0.0098}_{-0.0067}$ (tot) and the slope $alpha^prime(t{=}0) = 0.233 {}^{+0.067 }_{-0.074 }$ (tot) GeV${}^{-2}$ of the $t$ dependence are extracted for the case $m_Y{=}m_p$.
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