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The effect of spin–rotation coupling is measured for the first time with neutrons. The coupling of spin with the angular velocity of a rotating spin turner can be observed as a phase shift in a neutron polarimeter set-up. After the neutron’s spin is rotated by 2 π through a rotating magnetic field, different phase shifts are induced for ‘up’ and ‘down’ spin eigenstates. This phase difference results in the rotation of the neutron’s spin-vector, which turns out to depend solely on the frequency of the rotation of the magnetic field. The experimental results agree well with the solutions acquired by the Pauli–Schrödinger equation.