We propose a new hypothesis for the origin of protoplanetary discs with large inner holes (or gaps), so-called transition discs. Our gas disc model takes into account layered accretion, in which poorly ionized low-viscosity dead zones are sandwiched by high-viscosity surface layers, and photoevaporative winds induced by X-rays from the central stars. We find that a gap opens at a radius outside a dead zone, if the mass-loss rate due to photoevaporative winds exceeds the mass accretion rate in the dead zone region. Since the dead zone survives even after the gap opens, mass accretion on to the central star continues for a long time. This model can reproduce large gap sizes and high mass accretion rates seen in observed transition discs.