We discuss the ellipticity of the single layer boundary integral operator for the wave equation in one space dimension. This result not only generalizes the well-known ellipticity of the energetic boundary integral formulation in L², but it also turns out to be a particular case of a recent result on the inf-sup stability of boundary integral operators for the wave equation. Instead of the time derivative in the energetic formulation, we use a modified Hilbert transformation, which allows us to stay in Sobolev spaces of the same order.