When a quantum state evolves in such a way as to describe a closed loop in the space of pure state density matrices, it must, as a consequence of the non trivial topology of this space acquire a path-dependent phase. When the state vector | ψ 〉 evolution is such that < ψ | d/dt | ψ > = 0, the resulting phase is that introduced by Aharanov and Anandan (thereafter called the A.A. phase). Mathematically this condition corresponds to a parallel transport of | ψ 〉 by a connection defined on a fiber bundle. This paper contains an elementary and self-contained discussion of the A.A. phase for a spin-1 system. In this case, the phase appears as the holonomy of the natural connection over the complex projective space P 2(C). Experimental verification of these ideas requires expressions for both the phase in terms of the path and a Hamiltonian which will parallely transports the state vector along the path. They are given in terms of four directly measurable quantities which parametrize the pure state spin-1 density matrices. It is not possible to measure directly the A.A. phase on an isolated system ; it requires the separating and subsequent bringing together of two subsystems which undergo different evolutions. We suggest two ways in which, in principle, the A.A. phase might be measured in the laboratory.