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	97	65	0	0.599	0.0
         Chicago	92	70	0	0.568	5.0
      Cincinnati	83	79	0	0.512	14.0
       St. Louis	78	84	0	0.481	19.0
      Pittsburgh	71	91	0	0.438	26.0
           Total	421	389	0

St. Louis can finish the season with at most 78 + 0 = 78 wins and will finish the season behind the following teams:
Milwaukee has already won 97 games.
Chicago has already won 92 games.
Cincinnati has already won 83 games.
# St. Louis Cardinals' magic number
# St. Louis's magic number
# Cardinals' magic number
# MLB Playoff Race