A full description of coexisting phases includes their respective compositions, which are provided by the thermodynamic tie-lines. Fluorescence microscopy enables visualization of coexisting lipid phases in giant unilamellar phases, but the composition information is missing. For cholesterol-containing lipid mixtures, knowledge of the compositions of the coexisting phases is important for understanding the nature of "membrane rafts". We propose and demonstrate a new method, based on ESR spectroscopy, for determining tie-lines in regions of two-phase coexistence in a ternary lipid mixture. Over 100 different lipid compositions containing the spin-labeled phospholipid 16-PC in or near the two-phase coexistence region of the liquid-disordered and the gel phases of dipalmitoyl-PC/dilauroyl-PC/cholesterol (DPPC/DLPC/Chol) were studied to determine five tie-lines, spread over virtually the full range of this coexistence region. The method is based on the facts that (1) along a tie-line the ESR spectrum must be a superposition of the two ESR spectra from the respective single phases at the phase boundaries (connected by the tie-line) in a ratio given by the lever rule; (2) along a tie-line the partition coefficient, K(p), for the spin-label, which is also determined in our method, must be constant. We do find that K(p) for 16-PC is close to unity, but its value depends on the particular tie-line. The coexisting phases in equilibrium are characterized by the K(p) of the spin-label and its respective dynamic parameters obtained from fitting the ESR spectra to dynamical models.