The detector of the E865-collaboration at the Brookhaven-AGS described here combines a magnetic spectrometer for the charged decay products of 6 GeV/c K+ with excellent electromagnetic calorimetry and efficient particle identification for electrons and muons. Its high-resolution, large acceptance and high rate capability made it well suited for the study of extremely rare or forbidden decays with multi-leptonic final states such as K+→π+μ+e−, K+→π+l+l−, K+→l+νle−e+ and K+→π+π−e+νe down to branching ratios below 10−11 in an intense K+ beam (≈108 per AGS spill).