Conversely, what about: {date} is in the last week of {month}?
Is the first week of April the first week that is entirely in April or is it the first week that contains the first of April? Of course this all changes when considering whether the start of the week is Sunday or Monday.
There may not even be an "answer". In that case I'm interested in the most common usage.
Edit:
This SO post and this mailing list are what have caused me to call the meaning into question in the first place.
Best Answer
In my (AmE) experience, the phrase is ambiguous and can mean any of the first week containing a date in April, the first week in which more days are in April than aren't, or the first week entirely contained in April, with the middle option being the most prevalent.
For example, if April 1 was a Friday or Saturday, in most circumstances, I would not expect the week of Mar 25/26-Apr 1/2 to be described as the first week in April.
Due to this ambiguity, this phrase is normally only used when giving rough estimates of dates where precision isn't necessary. If you want to precisely specify the range, use the actual dates.