Precise measurements of the time-dependent intensity of the low-energy (<50 GeV) galactic cosmic rays (GCRs) are fundamental to test and improve the models that describe their propagation inside the heliosphere. In particular, data spanning different solar activity periods, i.e., from minimum to maximum, are needed to achieve comprehensive understanding of such physical phenomena. The minimum phase between solar cycles 23 and 24 was peculiarly long, extending up to the beginning of 2010 and followed by the maximum phase, reached during early 2014. In this Letter, we present proton differential spectra measured from 2010 January to 2014 February by the PAMELA experiment. For the first time the GCR proton intensity was studied over a wide energy range (0.08–50 GeV) by a single apparatus from a minimum to a maximum period of solar activity. The large statistics allowed the time variation to be investigated on a nearly monthly basis. Data were compared and interpreted in the context of a state-of-the-art three-dimensional model describing the GCRs propagation through the heliosphere.