Here’s an interesting exchange from Zenit recently, answered by Father Edward McNamara. He was asked by a deacon about whether he can baptize during Mass. “My current pastor believes that a deacon baptizing during Mass would constitute a change of presiders,” the deacon wrote. “I can find nothing to support this view. Can you enlighten me?”
He replied that while there is nothing in writing about deacons doing baptisms during Mass, there is a document from the Holy See — a private response, not official instruction — regarding marriage, stating that only the celebrating priest can witness a marriage and receive the vows during a wedding Mass. Having another priest or a deacon do that would constitute a change of presider.
Father McNamara goes on:
The same principle of no change in the presiding celebrant would also apply to other liturgical celebrations unless the rubrics specifically allow for the direct participation of other priests or deacons without, strictly speaking, implying a change in the presiding celebrant.
This is foreseen, for example, in a concelebration in which other priests may recite alone a part of the Eucharistic Prayer. It is also possible to divide some parts of the rite of the anointing of the sick.
Also, the rites of baptism foresee the possibility, when the number of children to be baptized is very large, that the priest celebrant may be assisted by other priests and deacons in some of the rites such as the anointings and the baptism itself. Analogously, the bishop may delegate priests to assist him in confirming large numbers of candidates. None of these practices constitute a change of presider.
Finally, with respect to celebrating baptism during Sunday Mass, the Rite of Baptism of Children does not advocate it as a regular or habitual practice, to wit:
“9. To bring out the paschal character of baptism, it is recommended that the sacrament be celebrated during the Easter Vigil or on Sunday when the Church commemorates the Lord’s resurrection. On Sunday, baptism may be celebrated even during Mass, so that the entire community may be present and the relationship between baptism and eucharist may be clearly seen, but this should not be done too often. Regulations for the celebration of baptism during the Easter Vigil or at Mass on Sunday will be set out later.”
However, the rite does suggest a clear preference for a communal celebration on a Sunday:
“32. If possible, baptism should take place on Sunday, the day on which the Church celebrates the paschal mystery. It should be conferred in a communal celebration for all the recently born children and in the presence of the faithful, or at least of relatives, friends, and neighbors, who are all to take an active part in the rite.”
Unlike baptism within Mass, a communal celebration such as the above can be presided over by a deacon as ordinary minister.
You can read the whole thing here.
At my parish, we do baptisms outside of Mass on the first Sunday of every month, usually with six or seven babies. (I’ve done as many as 14, though.) I’m in rotation with two other priests to do them. Once in a while, people request a private baptism, which we discourage — primarily because it gets too complicated trying to coordinate the schedule of the minister (usually me), the family and the availability of the church.