We propose high-resolution laser spectroscopy of the 3.5 eV nuclear transition in Th-229 in isolated atoms. Laser excitation of the nucleus can be detected efficiently in a double-resonance method by probing the hyperfine structure of a transition in the electron shell. It is shown that for a suitably chosen electronic level, the frequency of the nuclear transition is independent of external magnetic fields to first order and of electric fields to second order. This makes Th-229 a possible reference for an optical clock of very high accuracy. The nuclear-electronic double-resonance method can be conveniently applied to a laser-cooled ion of 229Th3+ in a radiofrequency trap. Further applications of nuclear laser spectroscopy are discussed.