  Using velocity profile measurements based on dynamic light scattering and
coupled to structural and rheological measurements in a Couette cell, we
present evidences for a shear-banding scenario in the shear flow of the onion
texture of a lyotropic lamellar phase. Time-averaged measurements clearly show
the presence of structural shear-banding in the vicinity of a shear-induced
transition, associated to the nucleation and growth of a highly sheared band in
the flow. Our experiments also reveal the presence of slip at the walls of the
Couette cell. Using a simple mechanical approach, we demonstrate that our data
confirms the classical assumption of the shear-banding picture, in which the
interface between bands lies at a given stress $\sigma^\star$. We also outline
the presence of large temporal fluctuations of the flow field, which are the
subject of the second part of this paper [Salmon {\it et al.}, submitted to
Phys. Rev. E].
