OK, this is a loaded question. The answer is, in my opinion, both. I won't get into the many, many years of theological debate re: predestination vs. free will (I've provided a few thoughts on the subject elsewhere). In general, we all have the God given ability to believe or have faith as a function of our will. Some would say our mind and will are tainted by sin and bend away from faith in Christ, thus God gives us His faith to believe (or chooses to make us righteous so we can believe thus the predestination angle). I take a slightly different approach…"And I, if I am lifted up from the earth, will draw all men to Myself" (John 12:32 NAS95S). I believe that God is drawing ALL men to Himself as a result of Christ's work on the cross. ALL men, however, must then choose to believe (and many will not). Some might argue that type of faith, as I have defined it, is a work of man and we all know "For by grace you have been saved through faith; and that not of yourselves, it is the gift of God; not as a result of works, so that no one may boast” (Eph 2:8-9 NAS95S). The sticking point here is that the word “gift” is not referencing "faith" but rather His grace and salvation. Salvation is not a work we do it is a gift of God that we receive by faith.

“being justified as a gift by His grace through the redemption which is in Christ. Jesus” (Rom 3:24 NAS95S)...note that the "justification" is the gift here as a result of the cross.

“For if by the transgression of the one, death reigned through the one, much more those who receive the abundance of grace and of the gift of righteousness will reign in life through the One, Jesus Christ.” (Rom 5:17 NAS95S)...note that "righteousness" is the gift that we "receive".

One other thought…God has also defined faith in the scriptures as choosing to believe. “But to the one who does not work, but believes in Him who justifies the ungodly, his faith is credited as righteousness” (Rom 4:5 NAS95S). "Believes in Him" is defined as "his faith" and further defined as "the one who does not work". Believing is not a work, it is called faith.

So, my faith or God's gift?…how 'bout BOTH

Absolutely not! Those who live by grace understand the truth of who they are in Christ (holy, pure and righteous)...sin is the opposite of their identity. You do not need a rule or a law to conform to your own identity...you are just you! And the "new you" created by God at salvation IS identified with the Lord Jesus Christ and IS by nature forgiven, righteous and accepted. Grace is not a license to sin but rather a license to LIVE as who you really are in Christ!

Repentance gets a lot of bad press. In my thinking it is grossly misused by the Church. The word simply means to think again, or consider after (reconsider); the context of scripture defines what we are to "re-think".

Many Christians attempt to define the word to mean an 180 degree turn from your sins...under the context of the believer, I might somewhat agree, but not for the unbeliever. The unbeliever does not turn from their sins to get saved, they simply reconsider the truth of the gospel and believe ((Mark 1:15 NASB) and saying, "The time is fulfilled, and the kingdom of God is at hand; repent and believe in the gospel."

Salvation is a free gift of God, anything we do (including "turning from our sins") outside of receiving His gift of grace is attempting to merit His favor...that's not salvation. The bottom line: are you trusting in the finished work of Christ on the cross or something else for your salvation. Consider again or rethink what/who you're trusting in and trust Christ alone...that's repentance unto life eternal.

Check out an excellent theological article on the subject at:
http://www.bible.org/docs/theology/soter/wilkin/wilkin-03.htm#TopOfPage

First, the answer to the question is no. Second, the real issue is, "do believers still have the capacity to sin"? The answer is, ...yes. 

As a believer, our essential nature is "spirit". That spirit is either alive "in Christ" or dead "in Adam". To have 2 natures would mean a person possesses 2 spirits...quite an impossibility since your essential essence is spirit...in other words, when you die your spirit (singular) is separated from the body ((James 2:26 NASB) "For just as the body without the spirit is dead...."). You are a spirit being who lives on this earth in an earth suit we call a body...that spirit has life in Christ or is separated from life in Christ (i.e. death).

The capacity to sin still remains in the body (Romans 7 - this does not make the body evil, it simply means this body and our mind is susceptible to sin until we receive our heavenly bodies) and satan will use the old patterns of believing and behaving to deceive us into making wrong choices.

(2 Cor 6:16 NASB) Or what agreement has the temple of God with idols? For we are the temple of the living God; just as God said, "I WILL DWELL IN THEM AND WALK AMONG THEM; AND I WILL BE THEIR GOD, AND THEY SHALL BE MY PEOPLE.
The believer has the Lord Jesus Christ living on the inside (i.e. in their spirit). Since the Lord can not be joined to evil, it is not possible for the nature of the believer to include evil (1 Cor 6:17 NASB But the one who joins himself to the Lord is one spirit with Him. (i.e. we are baptized in the spirit. We are not God, but are joined together with Him)).

The only way you could lose your salvation would be if you earned it in the first place...and you didn't, you received the free gift of salvation by faith. The scripture says, "In Him, you also, after listening to the message of truth, the gospel of your salvation-- having also believed, you were sealed in Him with the Holy Spirit of promise, who is given as a pledge of our inheritance, with a view to the redemption of God's own possession, to the praise of His glory (Eph 1:13-14 NASB.

“and I give eternal life to them, and they will never perish; and no one will snatch them out of My hand.”
(John 10:28 NAS95)

Salvation is a one time event of you receiving His free gift of Life in Christ Jesus...it does not depend on your continual faithfulness (even in believing) but rather on His faithfulness. "If we are faithless, He remains faithful; for He cannot deny Himself." (2 Tim 2:13 NASB)