We study the prospect of explaining the outbursts of Soft X-ray Transients (SXTs) by the thermal-viscous instability in a thin disc. This instability is linked to hydrogen ionization and is significantly changed when irradiation of the disc by X-rays from the inner regions is included. We present the first numerically reliable, physically consistent calculations of the outburst cycles which include the effects of accretion disc irradiation. The decay from outburst is governed by irradiation, as pointed out by King & Ritter (1998), leading to slow exponential decays. At the end of the outburst, the disc is severely depleted, which lengthens the time needed to rebuild mass to the critical density for an outburst. Despite this, the long recurrence times and quiescent X-ray luminosities of SXTs still require the inner disc to be replaced by another type of flow in quiescence, presumably a hot advection dominated accretion flow (ADAF). We include the effects of such truncation and show that the resulting lightcurves are conclusively similar to those of SXTs like A0620-00. We conclude that the two-alpha disc instability model provides an adequate description of the outbursts of SXTs when both truncation and irradiation are included. The values for the viscosities in outburst and in quiescence are comparable to those used in CVs. We discuss the model in the context of present observations.