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	93	69	0	0.574	0.0
         Chicago	83	79	0	0.512	10.0
       St. Louis	83	79	0	0.512	10.0
      Cincinnati	77	85	0	0.475	16.0
      Pittsburgh	76	86	0	0.469	17.0
           Total	412	398	0

St. Louis can finish the season with at most 83 + 0 = 83 wins and will finish the season behind the following teams:
Milwaukee has already won 93 games.
# St. Louis Cardinals' magic number
# St. Louis's magic number
# Cardinals' magic number
# MLB Playoff Race