Marriage is a legal, binding contract, recognized and registered with the state.
Marriage is also a sacred union, recognized by the Catholic Church.
The legal wheels can turn fairly quickly, enabling the contract to be declared void. Upon the issuance of the divorce decree, the state declares the marriage to be at an end, and both parties are free to pursue their lives, even to entering into another marriage contrat, recognized by the state.
The Catholic Church would view things a little differently.
Divorce is a sin, under canon law. But the Church does recognize the process of annulment (fundamentally declaring that the marriage never existed in the first place).
Technically a Catholic would have to receive an annulment from the Church (a religious nicety that the state doesn't recognize), as well as a divorce from the state (a legal action that the Church doesn't recognize). Technically, the Church recognizes the couple as still married, until the annullment is granted, even if they are divorced.
Re-marrying, before receiving the annulment is a sin, but not a crime.
Technically speaking the Catholic church is NOT following the Bible. At some point you must decide, WHO are YOU going to follow? Perfection or a fallible mortal?
Just pointing out the obvious minus the unbiblical rationalizations.