Heterogeneities in the cell membrane due to coexisting lipid phases have been conjectured to play a major functional role in cell signaling and membrane trafficking. Thereby the material properties of multiphase systems, such as the line tension and the bending moduli, are crucially involved in the kinetics and the asymptotic behavior of phase separation. In this Letter we present a combined analytical and experimental approach to determine the properties of phase-separated vesicle systems. First we develop an analytical model for the vesicle shape of weakly budded biphasic vesicles. Subsequently experimental data on vesicle shape and membrane fluctuations are taken and compared to the model. The parameters obtained set limits for the size and stability of nanodomains in the plasma membrane of living cells.