Originally posted by Greadius
The French do it in their presidential election 
They just have a week between the two so we pretend its one big election, as compared to the decentralized primaries but open presidential ballot.
Are you saying that the first round is meaningless, and that only the second one counts, which would mean that the small candidates cannot take part in the vote that matters? Because if you do, you are utterly wrong.
Quick reminder for people not familiar with the French presidential elections:
-First, there is a vote between all the candidates, small one included. Everyone cast one vote for the candidate of his choice.
-Second, there is a "run-off" two weeks later between the candidates who got the most votes in the first election.
Anyway, since France is not a two-party state, who gets to the second round is not at all a forgone conclusion. In 1995, while it was about certain that the leading left)wing candidate (Jospin) would make it, the right-wing vote was split between Chirac (Gaullist) and Balladur (Centrist). Chirac ended second and went on to defeat Jospin in the second vote.
More dramaticaly, in 2002 the votes of both the mainstream right & left where split between a multitude of parties, which led to the terrible result of the far-right candidate getting a bit more voice than the mainstream left one (Jospin again) and going to the second vote to face Chirac (and lose 82% - 18%).
So the first round does matter...
What is more, usualy it enable small parties normaly allied with the main left or right-wing parties to present an independant candidate, and to mesure their electoral strength that way, while having their elector vote in the second round for the "big" candidate they want to win.
With such a system in place in the US, it would allow a Green Party member to vote Nader in the first round, and still get to vote for the Democratic candidate in the second.
On the whole, it's a good system, which allow small parties to compete by avoiding the "vote-wasted" problem, and makes sure the candidate who wins is picked by half the electors, at least by default.
It also mean that when a non-democatic (as in "in favour of democracy") candidate manages to get though to the second round with only 16% of the vote due to vote division, the rest of the electorate can vote massively for the other candidate.