Orbital angular momentum (OAM) of light is an attractive degree of freedom for fundamental studies in quantum mechanics. In addition, the discrete unbounded state-space of OAM has been used to enhance classical and quantum communications. Unambiguous measurement of OAM is a key part of all such experiments. However, state-of-the-art methods for separating single photons carrying a large number of different OAM values are limited to a theoretical separation efficiency of about 77%. Here we demonstrate a method which uses a series of unitary optical transformations to enable the measurement of light's OAM with an experimental separation efficiency of >92%. Furthermore, we demonstrate the separation of modes in the angular position basis, which is mutually unbiased with respect to the OAM basis. The high degree of certainty achieved by our method makes it particularly attractive for enhancing the information capacity of multi-level quantum cryptography systems.