We prove that the norm of the Euler class E for flat vector bundles is 2^–n (in even dimension n, since it vanishes in odd dimension). This shows that the Sullivan–Smillie bound considered by Gromov and Ivanov–Turaev is sharp. In the course of the proof, we construct a new cocycle representing E and taking only the two values ±2^–n. Furthermore, we establish the uniqueness of a canonical bounded Euler class.