The Church of England is entering "a completely new phase of our existence" after the appointment of women bishops was approved in a historic vote, the Archbishop of Canterbury said.
The Most Rev Justin Welby said that women could make up half of all bishops within 10 to 15 years after the General Synod passed a change to ecclesiastical law giving them the same right to fill the post as men.
The vote in London, which was met with silence and followed by prayers, comes 20 years after the ordination of the first women priests in the Church of England. It is thought that the first woman could be appointed as a bishop next year.
Archbishop Welby told reporters: "It has taken a very, very long time and the way is now open to select people for the episcopacy, to nominate them on the basis simply of our sense that they are called by God to be in that position without qualification as to their gender."
He added: "We are working very, very hard on training and development of people, men and women, for senior posts in the church.
"The aim is ... that you end up with a big pool of people where gender is irrelevant and that that pool is pretty evenly mixed.
"There is a very conscious effort to make it easy for Crown Nomination Commissions to be able to have a fair choice between men and women. And we are going to take this very very seriously."
The change passed today saw the simple addition of a sentence to Canon 33, which reads: "A man or a woman may be consecrated to the office of bishop."
Only a few Synod members voted against it, including several women.
The Very Rev Dr Jane Hedges, 58, dean of Norwich, is the bookies' favourite to become the first woman bishop. Ladbrokes has offered 3/1 for the former canon steward of Westminster Abbey and archdeacon of Westminster.
She leads the Ven Rachel Treweek, archdeacon of Hackney (6/1), and the Very Rev Dr June Osborne, dean of Salisbury Cathedral (8/1).
Other leading candidates include the Very Rev Vivienne Faull, dean of York Minster.
The first diocese vacancy to come up after the "canon" law is changed will be Southwell and Nottingham, after the Rt Rev Paul Butler was appointed as Bishop of Durham.
It will be followed by Gloucester, Oxford and Newcastle.
The General Synod overwhelmingly backed legislation introducing the first women bishops in the Church of England in July and today's vote rubber-stamped the move.
It was given Royal Assent by the Queen, the spiritual head of the Church of England, last month.
Synod member Christina Rees, who has campaigned for women in the church and female bishops for almost 25 years, said it was a "truly historic moment" but said it was a pity that it was received in "dignified silence".
She said: "I thought today we might actually manage a bit of a cheer and applause for what, after all, the church has been asking for for a number of years.
"We know the vast majority of people in the Church of England just simply can't understand why it has taken so long.
"But we are here, we have done it, it is very good news."
She said some diocese would have to take the step of appointing a woman instead of a male "safe pair of hands", adding: "I think what the church needs to understand, and I think a lot of people do understand, is that now there are a number of very senior women who will have had more experience in the Church of England at a senior level than any of the men have, because they have been kept under this stained glass ceiling, which today was shattered."
One Synod member urged the archbishops to use their "formidable powers" to ensure that the first women bishops are appointed quickly.
Tim Allen, from the Edmundsbury and Ipswich diocese, called for "the appointment of as many as possible of the best of the Church of England's excellent senior women as diocesan bishops, preferably with seats in the House of Lords".
He also said that openly gay men were the next group who should be treated equally, saying: "It is not only women who are excluded in a discriminatory and prejudiced way from the house of bishops. So too were, and still are, those gay men who do not hide their sexuality in the closet.
"Those who are honest and frank enough to live openly in a civil partnership, while behaving in the chaste way demanded by church law are, it seems, from all the evidence, de facto excluded from the house of bishops, even when they are eminently qualified to be a bishop."
Bishop of Salisbury the Right Rev Nicholas Holtam said: "It was a great joy to go to General Synod to vote on the ordination of women to the episcopate for the last time. Alleluia!
"Last night, in Salisbury Cathedral, there was a service to mark 20 years of women in priestly ministry. We looked back with thanksgiving and forward with hope. For women to minister alongside men in all three orders of ordained ministry is, in George Herbert's phrase, 'heaven in ordinary' - and, for the C of E, 'something understood'."