All prime ideals of the enveloping algebraU=U(sl3), over an arbitrary field of characteristic zero, are listed by generators and classified by height. The inclusions between these ideals are also determined. Each prime idealIofUis presented by generators in several different ways. At one pole, we give a generating set forIas an adjoint module, and this includes a description for the expression of every possible highest weight element ofI. At the other pole, except for the case when the ideal is generated by the central variablesIhas a single generator as a two-sided ideal ofU.