Hi Andrew,
I do not know the US system well but I think that to do a PhD, the grant award linked above is not your best port of call. It is almost guaranteed that they will only award it to a PI with years of postdoctoral experience (in all probability, if we’re honest, someone who is already known to a decision-maker). Those awards are not, I think, intended to fund PhD fellowships.
Depending on which year you are in now, I suggest you have a look e.g. at this:
http://scgf.orau.gov/
It is closed for this year but I’m not sure if you were intending to start a PhD this fall or not.
See also
http://www.profellow.com/fellowships/how-to-fully-fund-your-phd/
http://www.hertzfoundation.org/dx/fellowships/eligibility.aspx
There must be many similar resources. Also maybe consider studying in a country where it is unusual for PhD students to pay to study e.g. with
http://www.fulbright.org.uk/fulbright-awards/exchanges-to-the-uk/postgraduates
Again they are looking for 2014-15 now. Hope you can find what you are looking for.