The author constructs a complete set of observables on the infinite-dimensional phase space of cylindrically symmetric gravitational fields. These observables have vanishing Poisson brackets with all constraint functions of the theory and are complete in the sense that any 'gauge invariant' function can, on the constraint surface, be expressed as a function of the observables. The key to the isolation of the observables is the observation that, in generally covariant theories, observables represent nothing more than a correlation of the true degrees of freedom with certian other phase space variables which represent time. Following the work of Kuchar (1989), the true degrees of freedom are identified with the free Cauchy data for the Einstein-Rosen waves, while the variables representing the (many-fingered) time are identified with the Einstein-Rosen 'extrinsic time' and space variables. The observables are constructed by solving for the evolution of the true degrees of freedom in terms of the many-fingered time and then inverting the solution to obtain the 'initial data' of the theory as functions on the gravitational phase space.