I often see it done, but I think it is wrong. This prayer is not included in the Roman Missal. Some people seem to think: “We compose the Prayer of the Faithful, so we are free to include the Hail Mary at this part of the Mass.” But there are instructions about the Prayer of the Faithful, which should be followed:
“71. … The people, however, stand and give expression to their prayer either by an invocation said together after each intention or by praying in silence.” (2002 General Instruction of the Roman Missal, GIRM). The Hail Mary does not fulfill this.
Singing the Hail Mary seems acceptable to me. From the GIRM: “139. When the Prayer of the Faithful is completed, all sit, and the Offertory chant begins (cf. above, no. 74).
An acolyte or other lay minister arranges the corporal, the purificator, the chalice, the pall, and the Missal upon the altar.”
So sit down and sing it. But do not remain standing to include the Hail Mary as part of the Prayer of the Faithful.