There was a baptism during the Mass that our (adult) choir sang at too. And it wasn’t the last Mass of the day either. (The grandma of the baby sings in the adult choir, however. I don’t know if that had anything to do with the choice of Mass time.)
I can’t for the life of me imagine a baptism during a Mass for emergancy purposes… (Unless some unbaptized person stops breathing, has a heart attack, or other life threatening event during Mass.) Baptisms during Mass usually involve coordination between priest, deacon, music director, ushers, readers, altar servers, etc.
I find it more likely that the priest only permits baptisms during Mass for special circumstances only. This is because it takes a bit more planning and possibly extends the length of the Mass. (Which could be an issue if another Mass will follow this one.) My guess is that ‘special cirumstances’ means one of two things: the child is from a signifcant family in the parish or the pastor thinks the sacrament of baptism is especially liturgically appropriate for a particular day or season.
As others have mentioned, the practice of having baptisms during Mass varies from diocese to diocese and even more so from parish to parish. It has been my experience that some people embrace the idea of baptisms (and other sacraments) during Mass and others think that baptisms are a big distraction. Pastors will have their own viewpoints on the matter and will likely be influenced by the viewpoints of parishioners.
In my parish we tend to have baptisms during Mass at the final Sunday Mass. And even then, we usually limit the number of children to two. Most baptisms are immediately after the last Mass. In either case it is usually a deacon who baptizes. Families are asked for their preference as to whether baptism will be during Mass or after and are usually accomodated.