The fractional integral of Weyl has not been so extensively studied as the one of Riemann-Liouville and only some of its properties have been proved with restrictive conditions. The fractional integral of Riemann- Liouville acts as a fractional power of a suitable integral operator in a very general class of functional spaces, such as the space of locally integrable functions in \([0,\infty[\). This is not the case for the fractional integral of Weyl, for which the ideal setting to study it is the Banach space \(L^ p(]0,\infty[)\). In this paper we study this fractional integral considering it as an operator defined on \(L^ p(]0,\infty[)\), and we describe it as a fractional power of a suitable integral operator and, consequently, we obtain some important properties for a rather wide class of functions, directly. We also establish a formula for fractional integration by parts which relates the fractional integral of Riemann-Liouville to that of Weyl.