We have searched for a deeply bound kaonic state by using the FINUDA spectrometer installed at the e(+)e(-) collider DAPhiNE. Almost monochromatic K(-)'s produced through the decay of phi(1020) mesons are used to observe K(-) absorption reactions stopped on very thin nuclear targets. Taking this unique advantage, we have succeeded to detect a kaon-bound state K(-)pp through its two-body decay into a Lambda hyperon and a proton. The binding energy and the decay width are determined from the invariant-mass distribution as 115(+6)(-5)(stat)(+3)(-4)(syst) MeV and 67(+14)(-11)(stat)(+2)(-3)(syst) MeV, respectively.