Under current law it is only an offence to incite suicide if the person attempts or commits suicide.
One of the major changes is a clause that will give web hosts two different ways of dealing with material posted on their sites.
They can either follow their own procedures or choose what is known as a "safe harbour" clause, which would offer them legal protection.
If online content hosts stick with the "safe harbour" option, which includes a 48 hour notice period before material is removed, they won't be held liable for any action they choose to take.
Opting out of the "safe harbour" option would not give rise to any legal liability for content hosts.
Both Labour and New Zealand First indicated they would support the bill through the committee stage after initially opposing it.
However Labour communications spokesperson Clare Curran said the party still had "grave" concerns about the bill.
Ms Curran said the bill would see cases referred to court that would waste the court's time and she was concerned about the lack of emphasis on education about cyberbullying.
She also submitted an amendment that would stop people under the age of 17 from being convicted under the new legislation.
When asked if Labour would support the bill at its third and final reading, Ms Curran said the party was "reluctantly" supporting the bill.
Communication Minister Amy Adams said real, significant harm could be caused through digital communications and it was important to address that.
She believed it was unlikely that young people would be convicted under the legislation unless there was no other avenue, as they would be dealt with in youth court.
New Zealand First MP Tracey Martin believed the minister's amendments had addressed the party's concerns about criminalisation and the party was now comfortable supporting the bill at its third reading.
