Once the couple confirms that they are prepared to get married, the actual marriage takes place. They join their right hands and declare the marriage publicly, by saying, "I, [Name], take you, [Name] to be my [wife/husband]. I promise to be faithful to you, in good times and in bad, in sickness and in health, to love you and to honor you all the days of my life."
Note that the couple bestow the sacrament on each other. It is not the priest who delivers the sacrament; the priest is there, as a symbolic witness of the Church. It is the two people who marry each other.
The marriage is accepted by the priest, who declares that "what God joins together, no one may put asunder." Then, the congregation accepts the marriage, in this format:
Priest: Let us bless the Lord.
All Respond: Thanks be to God.
Unscheduled applause usually takes place after this, because the couple is now married.
After this, the rubrics say "Another acclamation may be sung or said." I am not 100% sure what this means, since the Rite of Marriage does not give examples. I'm guessing it means you can do a short song, something like this:
Tune: Ode to Joy
God we praise you, God we thank you, in the heavens high above,
For the sacrament of marriage, symbol of your divine love.
Anything much longer than that, and you're looking a full-blown song, not an acclamation.