The amendment would offer Westminster Abbey the same kinds of protections against protests currently enjoyed by the Houses of Parliament.
Speaking about the amendment to the Bill, Bishop John Packer said: "The Abbey has suffered... as a result of what is often quite unintentional disturbance of its worship, and its role as a place of prayer and the worship of God."
While affirming "absolutely both the right of protest [and] the human rights which are involved in being able to protest", Bishop Packer maintained specific protections were necessary for the Palace of Westminster.
The new provisions would mean that those wanting to protest around the Palace of Westminster would need to give advance notice to local authorities.
If a protest were to emerge that had not been pre-cleared with the police, a fine would possibly be incurred.
Lord Deben, the Peer who proposed the amendment said "there was "an even greater issue for those who were trying to have a service within Westminster Abbey".
"Most of those who would be demonstrating would be very upset if they realised that their noise meant ... that the wedding day of someone who had looked forward to it, either in St Margaret's or in Westminster Abbey, would be destroyed because it would be impossible to hear.
"A number of those who demonstrate have strong religious views themselves and would not want that."
Baroness Berridge also gave her support to the amendment: "We have done much hard work to ensure that the work of the House of Commons is not disturbed; we should afford the same privileges to the Abbey and St Margaret's, which are in this unique position.
"It is not just that there is worship there and wedding services ... The young people being educated there and the people living there are... disturbed by the protests.
"This is a reasonable accommodation of the right to protest and the freedom of worship, while allowing people in their residential and educational roles to be uninhibited."
While the amendment debated in the House of Lords this week has gained support, the Bill more widely has received substantial public criticism, in particular over its provisions around protesting.
Netpol, the Network for Police Monitoring has said that the law would enable "police to disperse people taking part in a lawful assembly and arrest those that did not comply".
Without any oversight, any police officer with a rank higher than inspector could order a group to disperse from a location for at least 48 hours, if they believe that failing to do so would permit "anti-social behaviour".
"The police already have extensive powers to disperse or contain where there is a real threat of violence or criminality," Netpol said.
They point out that "Section 27 of the Violent Crime Reduction Act, also currently in force, allows police to disperse where they perceive a likelihood of alcohol related disorder" and that this law has already been criticised for ease of abuse by football fans.
"There is no need for these extended powers, which can only result in further restriction of freedom of assembly, and will increase the potential for discriminatory behaviour that will inflame community tensions."
Significant parts of this bill, specifically those relating to IPNAs - injunctions to prevent nuisance and annoyance that would have replaced the ASBO - have already resulted in House of Lords defeats.
An official e-petition demanding the bill's withdrawal has currently collected 11,465 signatures.
In its text, the petition appeals to the fact that "Organisations such as the Manifesto Club and the human rights group Liberty have expressed serious concerns regarding the Anti-Social Behaviour, Crime and Policing Bill."