I think you're misunderstanding a few of the issues; so let's get some background out of the way.
The First Vatican Council was an ecumenical council of bishops of the Catholic Church held at the Vatican in 1870. A number of important issues were discussed, and some Church doctrine was formalized; perhaps the most important of these issues was papal infallibility. (Not all Catholics agreed with what the Church had stated about the Pope being, in certain circumstances, infallible; many of these broke away and established what's now called the Old Catholic Church.)
The Second Vatican Council, another ecumenical council, was likewise held at the Vatican between 1962 and 1965. A number of things were discussed here, largely relating to the Church in its relationship to the modern world. Several documents came out of the Council: many appeared, at least at first sight, to contradict things which had been accepted for years in the Church, even stated as dogma. Answers to this question offer an overview of what the disagreements were and why they arose.
Unfortunately, neither those who accept not those who reject the teachings promulgated by Vatican II regard those on the other side as part of the true Catholic Church. It is reasonable to say, however, that far fewer than half a billion people (that is, half the number of people calling themselves Catholic) do in fact reject the Council and its teachings.
