The intrinsically disordered protein tau associates with microtubules in neurons but aggregates into cross-β amyloid fibrils that propagate in neurodegenerative brains. Different tauopathies have different structures for the rigid fibril core. To understand the molecular basis of tau assembly into different polymorphs, here we use solid-state nuclear magnetic resonance (NMR) spectroscopy to determine the structure of a tau protein that includes all microtubule-binding repeats and a proline-rich domain. This P2R tau assembles into well-ordered filaments when induced by heparin. Two- and three-dimensional NMR spectra indicate that R2 and R3 repeats constitute the rigid β-sheet core of the fibril. Unexpectedly, the amino-terminal half of R2 forms a β-arch at ambient temperature (24°C) but a continuous β-strand at 12°C, which dimerizes with the R2 of another protofilament. This temperature-dependent structure indicates that R2 is conformationally more plastic than the R3 domain. The distinct conformational stabilities of different microtubule-binding repeats give insight into the energy landscape of tau fibril formation.