OBSERVATIONAL estimates suggest that the cosmological density parameter Ω, defined as the ratio of the density of the Universe ρo and the critical density ρcrit which divides open from closed universes, is <=0.5. According to the standard Friedmann models, any value of Ω¬1 diverges away from one as the Universe evolves. The cosmological Ω-problem is that, on the one hand, making the present value of Ω close to but not equal to one requires an extraordinarily precise adjustment of the initial conditions of the Universe1, but on the other hand inflationary models, which were devised in part to avoid this fine-tuning, predict that Ω should differ from one only by an exponentially small amount2. Conventional inflationary models thus appeal to the existence of 'missing' matter or a non-zero cosmological constant to make up the density deficit. Here it is shown that 'extended inflation'3,4, a recent variation on inflationary cosmology, accommodates a range of initial conditions which lead to Ω<~0.5. The parameter range is narrow, perhaps finely tuned, but non-zero.