We report the measurement of the charmed baryonic decay B(0)-->Lambda(+)(c)p with a branching fraction of (2.19(+0.56)(-0.49)+/-0.32+/-0.57)x10(-5) and a statistical significance of 5.8 sigma. The errors are statistical, systematic, and the error of the Lambda(+)(c)-->pK(-)pi(+) decay branching fraction. This is the first observation of a two-body baryonic B decay. The analysis is based on 78.2 fb(-1) of data accumulated at the Upsilon(4S) resonance with the Belle detector at the KEKB asymmetric e(+)e(-) collider.