The Denmark-America Foundation (DAF), together with the Fulbright Commission, hold the largest pool of grants for PhD students going to the USA as part of their studies. 

The grants are at 85.000 DKK - 100.000 DK.

The deadline for applications is Wednesday, the 6th of March 2013 for students traveling within the 2013/14 academic year.

More information about the grants, as well as the application form, can be found here (in Danish only).