There is always the problem when answering this question that General Relativity, naively interpreted, allows you to speak about the part of the universe which is not observable from our vantage point, and this makes the question nontrivial.
But in a logical positivist perspective, the one suggest strongly by string-theoretic holography, the universe is exactly the stuff inside the cosmological horizon, and it is finite because the cosmological horizon is of finite area. There is no objective meaning to stuff outside the cosmological horizon, so there is no point in thinking about this--- it is meaningless in the sense of Carnap.