The zero-temperature system of two dilute overlapping Bose-Einstein condensates is unstable against long wavelength excitations if the interaction strength between the distinguishable bosons exceeds the geometric mean of the like-boson interaction strengths. If the condensates attract each other, the instability is similar to the instability of the negative scattering length condensates. If the condensates repel, they separate spatially into condensates of equal pressure. We estimate the boundary size, surface tension and energy of the phase separated condensate system and we discuss the implications for double condensates in atomic traps. PACS numbers(s):03.75.Fi, 05.30.Jp, 32.80Pj, 67.90.+z Typeset using REVTEX 1 As dilute gases, the atomic trap Bose-Einstein condensates  occupy a unique position among the superfluid systems. One intriguing consequence of their dilute gas nature is the prospect of studying condensate mixtures, which are the first experimentally realizabl
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