We investigate the possibility that the observed flatness of the rotation curves of spiral galaxies is not evidence for the existence of dark matter haloes, but rather a signal of the breakdown of General Relativity. To this aim, we consider power-law fourth-order theories of gravity obtained by replacing the scalar curvature R with f(R) =f0Rn in the gravity Lagrangian. We show that, in the low energy limit, the gravitational potential generated by a point-like source may be written as Φ(r) ∝r−1[1 + (r/rc)β] with β a function of the slope n of the gravity Lagrangian and rc a scalelength depending on the gravitating system properties. In order to apply the model to realistic systems, we compute the modified potential and the rotation curve for spherically symmetric and for thin disc mass distributions. It turns out that the potential is still asymptotically decreasing, but the corrected rotation curve, although not flat, is higher than the Newtonian one, thus offering the possibility to fit rotation curves without dark matter. To test the viability of the model, we consider a sample of 15 low surface brightness galaxies with combined H i and Hα measurements of the rotation curve extending in the putative dark matter dominated region. We find a very good agreement between the theoretical rotation curve and the data using only stellar disc and interstellar gas when the slope n of the gravity Lagrangian is set to the value n= 3.5 (giving β= 0.817) obtained by fitting the Type Ia supernova Hubble diagram with the assumed power-law f(R) model and no dark matter. The excellent agreement between theoretical and observed rotation curves and the values of the stellar mass-to-light ratios in agreement with the predictions of population synthesis models make us confident that Rn gravity may represent a good candidate to solve both the dark energy problem on cosmological scales and the dark matter one on galactic scales with the same value of the slope n of the higher-order gravity Lagrangian.