"If you love your neighbor, you will fulfill all the requirements of God's law. For the commandments, "Do not commit adultery," "Do not murder," "Do not steal," "Do not covet," and whatever other commandment there may be, are all summed up in this one commandment: "Love your neighbor as yourself." Love does no harm to its neighbor. Therefore love is the fulfilling of the law." {Romans 13:8-10 NIV, NLT}
It seems pretty clear in that passage that when Paul says, "love is the fulfilling of the law," that he's not saying love takes the place of keeping the law, but that love will lead us, or enable us, to keep the law.
He makes pretty much the same point in chapter 3, only there it's faith that's emphasized as enabling us to keep the law:
"Well then, if we emphasize faith, does that mean we can forget about the law? Of course not! In fact, only when we have faith do we truly fulfill the law." {Romans 3:31 NLT}