I don't know where George A. Darne is buried, but since Jefferson Barracks National Cemetery didn't have his remains, try the Confederate cemeter at Alton, IL. He may have been sent there and it not recorded on the records, but I think that is a stretch.
I think it is more likely that he was sent to one of a number of St. Louis area hospitals when he became ill, and his body was buried in some civilian cemetery where that medical facility normally buried their patients who died. W. B. Hesseltine's old article "Military Prisons of St. Louis, 1861-1865," in the April 1929 issue of "Missouri Historical Review" listed a number of health facilities that treated very sick POWs. The list on page 384 says these included "New House of Refuge, the City Hospital on Fifth Street, military hospitals on Fourth Street, Hickory Street, and at Jefferson Barracks, and the Sisters of Charity." Hesseltine's source for this list is a 28 June 1862 report to the national Provost-Marshal-General's office from the MO Provost-Marshal-General cited in "O.R." series 2, vol. 4, pp. 95-6.
Knowing which medical facilities treated sick POWs, however, will not by itself reveal where they buried their dead, so this is a dubious lead for you, but a lead just the same. I know it is dubious because most of those street names changed since the Civil War. If you wish to pursue this further in those directions, clues to deciphering the old street names of that neighborhood are in some old maps and some places in the text of William C. Winter's 1994 "The Civil War in St. Louis: A Guided Tour" published by the Missouri Historical Society Press in St. Louis (and available to you via interlibrary loan at your library).
If you know what date George Darne died in St. Louis you may wish to study the microfilm records of the two St. Louis daily papers, "Daily Missouri Democrat" and "Daily Missouri Republican" in the "Prison Hospital" column to see what details that source gives about his date and place of death. Even if you don't know his date of death, you could search these daily newspapers and read that column from about August 1863 on to see if you may happen to catch this notice of his death. You may order these microfilms from the Missouri State Historical Society from your local library via interlibrary loan.
Now that you have survived the above paragraphs I thought you may want to hear my theory as to how George A. Darne came to be arrested on 12 July 1863. I found your ancestor listed in Joanne Eakin's landmark, self-published 1995 book listing many of the POWs kept in the St. Louis area prisons. Eakin's listing, quoting from the Union clerk's scribbled entries, tells us that A. George Darne, citizen of Calloway County, was arrested there 12 July 1863 and sent to the St. Louis area military prisons. Sadly, that is all it says.
So I checked to see what was happening in Calloway County the few days just before July 12. I discovered that somewhere in Callaway County July 7 Union troops evidently of 1st Provisional Enrolled Missouri Militia captured three local guerrillas: John Day, W. Sing Duncan, and Charley Turner, who had all formerly been members of Rebel Captain J. D. W. Robinson's command. I could find no other details of the capture. I DID find, however, that POW John Day made the news the following month by his dramatic escape attempt from St. Louis' Gratiot Street Prison in which he was wounded (not dangerously) by four rifle bullets fired by the guards. While Day was being taken away for medical treatment he gamely commented to his captors "that sheet lightning couldn't kill him." He was referred to as a "noted bushwhacker," and with an attitude like that you can see why. There is an entry in Eakin's "MO POWs" for the indestructable John Day, too.
My theory about Darne's arrest and imprisonment is that he was rounded up as "one of the usual suspects" and sent off to prison for violation of oath and bond. Perhaps the Union militia in Callaway County had evidence that he had fed and aided the three captured guerrillas and their buddies. Maybe the Yanks found those three guys on Darne's property. Anyway, that is my theory since we don't have anything else.
My resources for the Callaway County July 1863 incidents and Day's ill-fated escape attempt in St. Louis include (besides Eakin's book):
--newspaper article "Missouri Items," "Daily Missouri Republican," St. Louis, 10 August 1863, quoted from earlier issue of the Fulton "Telegraph," date not given;
--newspaper article "Bushwhacker Caged," "Missouri Statesman," Columbia, 28 August 1863, quoting earlier "Saint Louis News," date not given.
I hope that helps you in your quest.