It is not possible to unambiguously distinguish between these states. To perform unambiguous state discrimination you need states orthogonal to all but one of the states to discriminate. So for example, here you'd need the measurement to contain an operator proportional to a state orthogonal to both $|0\rangle$ and $|1\rangle$. This is clearly not possible, as there's no such state.
This means for any possible measurement, for each observed outcome, you always have some ambiguity about the possible input state that caused it.
I should say that this is however different than asking whether state discrimination is possible "with a single measurement". Even when unambiguous state discrimination is possible, a single measurement outcome is still in general not sufficient to tell which state was used.