sadly, the withTimeAtStartOfDay answer is not correct, but only sometimes. You want:
It directs out that "midnight/start of day" sometimes involves 1 am (daylight savings appear this way in some areas), which Days.daysBetween doesn't hand accurately.
// 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 through a LocalDate sidesteps the entire problem.