Axions in the microeV mass range are a plausible cold dark-matter candidate and may be detected by their conversion into microwave photons in a resonant cavity immersed in a static magnetic field. We report the first result from such an axion search using a superconducting first-stage amplifier (SQUID) replacing a conventional GaAs field-effect transistor amplifier. This experiment excludes KSVZ dark-matter axions with masses between 3.3 microeV and 3.53 microeV and sets the stage for a definitive axion search utilizing near quantum-limited SQUID amplifiers.