Hello Michele,
Based on the info you gave that might indeed be hard. You could try the National Personell Records Center. See this link for the address and maybe some more addresses that might be helpful.
http://www.armyairforces.com/addresses.asp This site is about the US Army Air Forces but some of these addresses will be helpfull for Army people too.
However, getting the required information might take longer than 1 month.
Do you, by any chance have pictures of your dad on which patches and/or insignia's might be seen? Or perhaps you have documents relating to his military service? Any might contain clues to the ID of his unit. If not directly, then perhaps indirectly.
Sorry I can't be of more help.
Stevin