If you are Catholic, you trust that the Church has been given the keys (authority) of the kingdom (Mt 16:18) and that it is guided by the Holy Spirit (Jn 16:13) in terms of teaching on faith and morals. Thus, if you are Catholic, the proof comes with Ineffabilis Deus which was the formal declaration of the Immaculate Conception. If you are seeking proof beyond that, you are really questioning the matter of authority. Did the Church have the authority to make such a proclamation? Catholics must answer yes and I believe that the Scriptures do back up that authority. We have three pillars of authority-- Sacred Tradition, Sacred Scripture and Living Magisterium.
The Scriptural texts having to do with the Immaculate Conception are not direct proofs. Much of it is typological with the pure ark of the Old Covenant being a type of Mary. “Hail full of grace” is a powerful verse because of the Greek word used which shows that she had the fullness of grace before the Lord was conceived in her womb. Still, when I look for “proof” of dogma, I look no further than the authority of the Church.