The birth of the bride, Maria Fassmann, happened in the foundling house, which, I believe, was part of the Styrian State Hospital (Landeskrankenhaus Graz). The Findelhaus was a place for unmarried women to give birth, and receive related medical care. Most of these mothers did not have the means to care for their babies. Therefore, the children were available for adoption, or places of foster care were tried to be found for them.
The LKH (Landeskrankenhaus) Graz was and is one of the largest hospitals in Europe. Since many births and deaths happened there, including births of many children born "legally" (i.e., in wedlock), a dedicated parish was created called St. Anton. Geographically, the LKH Graz was part of the parish of St. Leonhard. There were periods of time when St. Anton played only a minor role, and St. Leonhard had the recording responsibility, or the father's home parishes in Graz were responsible.
In the marriage record, it is not clear whether the reference to St. Anton and St. Leonhard belongs to the groom. In any case, the birth occurred in the Findelhaus which in most cases meant the parish of St. Anton, possibly St. Leonhard. You should write to the Diözesanarchiv Graz and ask for a search and full transcription of Maria's birth/baptismal record. State that the birth was in the Findelhaus. State also that the original birth record should be located so that any subsequent marginal note, such as on her death, would not be overlooked.
Eric, write also to the State Archive of Styria (Steiermärkisches Landesarchiv) and ask for locating Maria's "Pflegschaftsakt" (file on the government's care provided to children who had no parent, or only the mother, to care for them). If the Pflegschaftsakt can be located, it would contain info about adoptive parents or foster parents and places for Maria. The file may even contain information about attempts by the authorities to determine the name of the biological father, and any alimony he had to provide for the girl.
Another place to find out more about Maria and her mother (Theresia Fassmann) would be the residency records ("Meldezettel") of Graz, and the census records of 1880, 1890, 1900 and 1910. These would be at the Archive of the City of Graz (Grazer Stadtarchiv). To a large extent, these records have been filmed by LDS. Searching through them is tedious, but highly rewarding. Another source you could inquire about at the Stadtarchiv are the midwife reports of births they assisted. These records contain data on the address of the women, and data normally not available through the official birth records, such as the number of childbirths preceeding the current one.
All the best for your search, regards,
Fritz Königshofer, Bethesda, Maryland, USA