According to his Civil War Pension Index card, William Ballard applied for, and received, a Civil War pension. His widow applied for, and apparently did not receive a pension.
You should order his pension records from NARA. You should order the full packet ($75), which will include all pages, up to 100, rather than the small packet ($25), which would include a maximum of 8 pgs.
These records should include a good amount of information. His date of death should be listed in these records.
I ordered the records for my Great-great-grandfather who served in the 133rd NY Vol Infantry and who died in La. in 1863. I received a ton of information, including his death record from the Army Reg. hospital, a copy of his marriage record, dates of birth of his 3 daughters and their baptismal dates - and lots more great information. I don't know how much you'll get, but hopefully you'll be reasonably lucky.