The Health Commissioner has organized two parks at the Insane Asylum grounds for the accomodation of inmates of the institution, one for the benefit of the males and the other for the females. The inmates are allowed two hours exercise in each. One of the features of the male department are two base ball clubs which have become very expert in the business. No trouble is experienced in getting the inmates back to their quarters, as at a designated time a signal is given and they file back into the building.-St. Louis Globe-Democrat, August 19, 1884
If I had to guess, I'd say this was the St. Louis County Insane Asylum, which opened in 1869. It could be St. Vincent's Asylum but the County asylum (which, in 1876, was renamed the St. Louis Insane Asylum) sat on something like forty acres, which is plenty of room for a baseball field.