According to an article in U.K.'s TimesOnline, researchers have found that women who chose home birth had triple the risk of stillbirth compared to hospital birth -- but only if they were classified as having high risk pregnancies.
I was concerned at first reading the headline of the article, which did not include the qualifier about high risk pregnancies. It seems to make sense that in high risk pregnancies it makes the most sense to have a hospital birth where there is a good amount of help readily available should complications arise, so that finding was not surprising.
I do think it should be reassuring, however, that the rates of stillbirth in home births for pregnancies not labeled high risk was not significantly higher than the rate of stillbirth in hospital births. Both my successful pregnancies were labeled high risk due to blood pressure issues (and the history of miscarriage I had acquired), so I never found myself seriously considering a home birth for myself although I could see a definite appeal for why women did choose them.