score:6
This is the question dealt with by the Sixth Ecumenical Council (the Third Council of Constantinople). The council addressed the heresy known as monothelitism. The heresy held that Jesus Christ only had one will.
The orthodox position was defined at that Council: Jesus Christ had two natures, human and divine; he also had two wills, human and divine. If there was no human will in Christ, the Incarnation would have been incomplete, as he would not have adopted everything that belongs to humanity (cf. Gregory of Nazianzus: "What is not assumed is not healed"). This is the logical outworking of Chalcedonian Christology (the idea that Christ is both human and divine).
The Catholic (and Orthodox, and Protestant) position is that the "not my will, but yours, be done" refers to the human will in Christ.
This is dealt with directly in the writings of Maximus the Confessor (I will try to find the citation), indirectly in the definition of the Sixth Council, and directly several times in the Summa Theologica of S. Thomas Aquinas (third part, question 18).