Why can't DateTime.ParseExact() parse "9/1/2009" using "M/d/yyyy"

I have a string that looks like this: "9/1/2009". I want to convert it to a DateTime object (using C#).

This works:

DateTime.Parse("9/1/2009", new CultureInfo("en-US"));

But I don't understand why this doesn't work:

DateTime.ParseExact("9/1/2009", "M/d/yyyy", null);

There's no word in the date (like "September"), and I know the specific format, so I'd rather use ParseExact (and I don't see why CultureInfo would be needed). But I keep getting the dreaded "String was not recognized as a valid DateTime" exception.

Thanks

A little follow up. Here are 3 approaches that work:

DateTime.ParseExact("9/1/2009", "M'/'d'/'yyyy", null);
DateTime.ParseExact("9/1/2009", "M/d/yyyy", CultureInfo.InvariantCulture);
DateTime.Parse("9/1/2009", new CultureInfo("en-US"));

And here are 3 that don't work:

DateTime.ParseExact("9/1/2009", "M/d/yyyy", CultureInfo.CurrentCulture);
DateTime.ParseExact("9/1/2009", "M/d/yyyy", new CultureInfo("en-US"));
DateTime.ParseExact("9/1/2009", "M/d/yyyy", null);

So, Parse() works with "en-US", but not ParseExact... Unexpected?