An apparently new expansion of the exponential integral E₁ in incomplete gamma functions is presented and shown to be a limiting case of a more general expansion given by Tricomi in 1950 without proof. This latter expansion is proved here by interpreting it as a "multiplication theorem". A companion result, not mentioned by Tricomi, holds for the complementary incomplete gamma function and can be applied to yield an expansion connecting E₁ of different arguments. A general method is described for converting a power series into an expansion in incomplete gamma functions. In a special case, this provides an alternative derivation of Tricomi's expansion. Numerical properties of the new expansion for E₁ are discussed.