We present a striking new Hubble Space Telescope (HST) observation of the rich cluster Abell 2218 taken with the Wide-Field and Planetary Camera (WFPC2). HST's restored image quality reveals a sizeable number of gravitationally-lensed features in this cluster, significantly more than had been identified using ground-based telescopes. The brightest arcs are resolved by HST and show internal features enabling us to identify multiply-imaged examples, confirming and improving the mass models of the cluster determined from ground-based observations. Although weak lensing has been detected statistically in this and other clusters from ground-based data, the superlative resolution of HST enables us to individually identify weakly distorted images more reliably than hitherto, with important consequences for their redshift determination. Using an improved mass model for the cluster calibrated with available spectroscopy for the brightest arcs, we demonstrate how inversion of the lensing model can be used to yield the redshift distribution of $\sim$80 faint arclets to $R\simeq25$. We present a new formalism for estimating the uncertainties in this inversion method and review prospects for interpreting our results and verifying the predicted redshifts.