Annoyingly, the withTimeAtStartOfDay answer is wrong, but only occasionally. You want:
Days.daysBetween(start.toLocalDate(), end.toLocalDate()).getDays()
It turns out that "midnight/start of day" sometimes means 1am (daylight savings happen this way in some places), which Days.daysBetween doesn't handle properly.
// 5am on the 20th to 1pm on the 21st, October 2013, Brazil
DateTimeZone BRAZIL = DateTimeZone.forID("America/Sao_Paulo");
DateTime start = new DateTime(2013, 10, 20, 5, 0, 0, BRAZIL);
DateTime end = new DateTime(2013, 10, 21, 13, 0, 0, BRAZIL);
System.out.println(daysBetween(start.withTimeAtStartOfDay(),
end.withTimeAtStartOfDay()).getDays());
// prints 0
System.out.println(daysBetween(start.toLocalDate(),
end.toLocalDate()).getDays());
// prints 1
Going via a LocalDate
sidesteps the whole issue.
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