The Archbishops put forward the amendment in a bid to stop opponents of women bishops leaving the Church of England.
There is strong opposition to women in the episcopate among conservative evangelical Anglicans and Anglo-Catholics, who believe the consecration of women as bishops goes against Scripture and Jesus' ordering of His church.
Under the amendment put forward by the Archbishops, the diocesan bishop – whether male or female – would have been legally entitled to exercise any Episcopal function in any parish of the diocese but in practice be forced to refrain from exercising certain functions in a parish that had requested the oversight of a male bishop.
The legal authority of the nominated bishop to minister would have derived from the Measure itself and would not have been conferred by way of delegation by the diocesan bishop.
Although the majority of General Synod members voted in favour of the amendment, it was lost after failing to secure a majority in all three Houses – Laity, Clergy and Bishops. The vote was carried 25 to 15 in the House of Bishops and 106 to 86 in the House of Laity, but defeated 85 to 90 in the House of Clergy, around one third of whom are women.
Although the Archbishops made clear that the vote on the amendment was not a vote of confidence in them, the defeat will be seen as an embarrassing blow to their authority.
The vote left Synod reeling, with some members taking to the stand to ask that the debate be adjourned immediately in order to give Synod members time to reflect and seek advice from colleagues.
The Rev Dr Jonathan Baker, a member of the Revision Committee that drafted the original Measure on women bishops, said the vote had left Synod in a "remarkable place in terms of the relationship between priests and bishops".
Leading the opposition to the amendment was Christina Rees of pro-women bishops group WATCH (Women and the Church). She told Synod the amendment would render women bishops "avoidable" and "legally optional", and create a two track system within the Church where one strand of episcopacy was restricted to men only.
"It is one thing to accommodate differences but quite another to contradict the very thing the law has been designed to enable," she said.
"Is it possible to have two concurrent jurisdictions in which one does not recognise the valid jurisdiction of the other?
"I do not see how we can reach out and make a difference to the world now and in the future unless we have men and women together exercising episcopal ministry and leadership in our church."
The Steering Committee of the Revision Committee also objected to the amendment, saying that it would make it unclear whether the jurisdiction of the diocesan bishop or the jurisdiction of the nominated bishop had priority.
Opponents of women bishops admitted their disappointment at the conclusion of the day long debate, which also saw the defeat of other amendments proposing the transfer of jurisdiction and the creation of separate dioceses.
The Rev Preb David Houlding, an Anglo-Catholic, said he did not want to leave the Church of England and that the Church should be big enough to accommodate his views.
Speaking in support of the Archbishops' amendment, he said: "Many years ago when there was in defeat in the air, we asked for bread and you gave us a stone. If you defeat this amendment now you won't have even given us a stone."
The Rev Rod Thomas, Chair of orthodox Anglican group Reform, said there was a "profound feeling of deflation" among opponents of women bishops following today's session.
He said the concerns of opponents had been "ignored" and that the defeat of the Archbishops' amendment had exposed a "loss of leadership" in the Church.
"There are big questions about the future given that the legislation that is likely to go forward for discussion in one or two years' time will be so obviously deficient that there are big questions about whether it will ever get final approval.
"By pressing so hard, the people who want to make clear provision for women bishops may have actually scuppered the whole thing."
Although Thomas expressed disappointment at the outcome of today's vote, he said he was not disappointed that roughly one third of Synod members had voted in favour of some kind of provision for opponents.
He added: "There is a good chance that at least one third will oppose it when it comes back round for debate again."
A spokesman for evangelical Anglican group Fulcrum, which supports women bishops, said it had been "unwise" of the Archbishops to put forward their amendment without first checking levels of support among interest groups.
"They left themselves open to defeat," he said.
He remained optimistic that women would eventually be elected to the episcopate, saying that there was a "broad core" of support in Synod.
"They're not going to leave this. It will come back and back and back," he said.
"It would be incredulous if it were delayed indefinitely. There is too much momentum behind it and there is too much of a justice issue behind it, in that it just doesn't make sense.
"If you have women as priests then women should be able to be in any other order of the Church."