This proposal, "Mono-Ion Research IV: Penning Trap Mass Spectrometer" is for the development of the world's most precise and accurate mass spectrometer. Using only rf methods for observing the normal mode frequencies of a single isolated ion, bound to a small electromagnetic cage called a Penning trap, it has become possible to measure masses of low mass-to-charge ions relative to a highly- charged carbon ion with a precision that now exceeds five parts in 1010 for a single day's run. The fundamental goal of such a high precision instrument is to increase the consistency and accuracy of the fundamental constants of nature. The immediate objectives will involve: using the accurate mass measurements to "weigh" chemical bonds, electron ionization energies, and nuclear excitation energies; improve the measurement accuracy of the electron's atomic mass by an order of magnitude; measure light-ion masses to one part in 1010 or higher accuracy and determine neutron separation energies of selected light ions by measuring consecutive isotopes of the given element; and refine the measurement of the tritium--helium-3 mass difference to an accuracy of 0.3 eV.