It seems keeping the Mosaic Law (in which the Ten Commandments resides) is no longer a requirement for Christians, since by Christ's life, death and resurrection, Christians are no longer under its bondage but under grace. Does this mean because Christians are under grace they get to do anything they want? Of course not! Galatians 5:1-6, 13-18, 22-25, 6:2, 2 Timothy 1:9-10, and Titus 2:11-13 gives a good explanation of what it means to live under grace.

It seems Paul is saying here that if one insisted on keeping the letter of the law (or the Old Covenant), Christ's sacrifice is of no use and all 613 laws (which include the Ten Commandments) found in the book of Moses must be kept, not just what one wishes to pick and choose to keep (e.g. circumcision, Sabbath days, etc.). As the verse 4 says, one in such a state is fallen from grace.

It seems living under grace instead of under the law means loving and serving one another, for Paul says in verse 14 that "the whole law" is fulfilled or summed up in loving one's neighbor as oneself. Living under grace is a call to walk in the Spirit, allowing the Spirit to transform us into the type of people God wants us to be. Accordingly, being led by the Spirit frees one from the law and produces some wonderful changes.

Galatians 6:2
"Bear one another's burdens, and thus fulfill the law of Christ."

Love as the central theme of the Gospel is also confirmed in Romans 13. Paul, similar to Christ in Matthew 7:12 (commonly called the Golden Rule), summed up the law in these words:

1 Timothy 1 provides an interesting insight into who the law is for. The law is not for those who are justified by faith by accepting the grace of Christ and His call for us to love one another as He loves us, but for those who live contrary to the Gospel, which is the law of Christ.