St. Louis has been eliminated from first place.

To show that St. Louis is eliminated from first place, we start with the current standings:
            Team	W	L	GL	PCT	GB
       Milwaukee	91	59	12	0.607	0.0
         Chicago	86	64	12	0.573	5.0
      Cincinnati	75	75	12	0.500	16.0
       St. Louis	73	78	11	0.483	18.5
      Pittsburgh	65	86	11	0.430	26.5
           Total	390	362	58

St. Louis can finish the season with at most 73 + 11 = 84 wins and will finish the season behind the following teams:
Milwaukee has already won 91 games.
Chicago has already won 86 games.
# St. Louis Cardinals' magic number
# St. Louis's magic number
# Cardinals' magic number
# MLB Playoff Race