Gary, try doing a Chester Street Prison, of simply a Chester Street search at:
http://www.sos.mo.gov/archives/provost/default.asp#search
You will get no results.
Try Chestnut Street, and you will get 13 results, such as this one fpr Chestnut Street Hospital:

"Roberts, Daniel; Nickell, Isaac R.; Lindsay, L. D.; Sherwood, William T.; Hancock, Columbus; Davis, J. J.; Moore, Thomas A. H.; Love, W. P.; Thomas, William; Reynolds, B. F.; Courts, R. H.; Wright, Benjamin; Humphreys, Benjamin; Berry, W. C.; etal
Saint Louis
Recommendations for release of prisoners from Chestnut Street Hospital and Fifth Street Military Prison; prisoner Moore from Saline County; two other prisoners listed, Jonathan W. Frazier and Henry Frazier
02-17-1862 F1583"

F1583 2-17-1862 can be viewed at:

http://www.sos.mo.gov/archives/provost/provostPDF.asp

Perhaps there is a clue to the name of the Prison in this?
Fran