Cosmic strings are topologically stable defects in the vacuum which may be produced at a phase transition in the early Universe. Some may form closed loops, and some may be open, extending over the entire observable Universe. Although gravitational lensing of distant quasars by cosmic strings has been discussed1-6 it is difficult to prove that a particular pair of quasar images is formed by a cosmic string. We suggest here that observations of very distant galaxies are more useful as a means for their discovery. A galaxy may appear as cut by a sharp edge if there is a cosmic string between the galaxy and the observer. We argue that if there is only one string out to a redshift z≈1, then the probability that it crosses a random image obtained with the Wide Field Camera (WFC) of the Space Telescope is ~10-4. Therefore, there is a fair chance of discovering a string, after taking 104 images. The WFC will operate almost continuously in primary and serendipity modes, and a cosmic string, if it exists, may be discovered using the Space Telescope within the first few years of its operation.