It means the proposal, backed by the Welsh Government, to call the institution both Senedd Cymru and Welsh Parliament, stands.
Senedd Cymru was proposed by Ynys Mon Plaid Cymru AM Rhun ap Iorwerth but it was defeated by 39 votes to 16.
AMs also confirmed that they would change their title to Members of the Senedd, or Aelodau o'r Senedd.
On Saturday around 100 people joined a protest backing the name Senedd, after more than 30 famous names - including actor Michael Sheen, singer Cerys Matthews and rugby referee Nigel Owens - signed a letter to AMs calling on them to Welsh-only name.
On Wednesday, Mr ap Iorwerth pleaded with the Welsh Government, made up of Labour AMs and one Liberal Democrat, to allow ministers to be given a free vote on the name.
"Senedd Cymru is Wales' Parliament," he said.
"We sing our national anthem in Welsh not to exclude anyone, but to include everyone, in a celebration of our Welsh nation in a way no-one else does theirs."
acking the name Senedd Cymru, Swansea East Labour AM Mike Hedges said it was "best to give it a name it's going to be called".
"The word Senedd is the word that's going to be used," he said.
He added: "In terms of bilingualism in Wales, I've never seen the word eisteddfod ever translated.
"People use the word eisteddfod although it happens to be a Welsh word."
Conservative AM David Melding said his group was convinced that a bilingual name for the institution was needed in the legislation "because the consultation to
the bill produced an emphatic response - over 70% - in favour of a bilingual name".
he Welsh Government's top legal adviser, Counsel General Jeremy Miles, said: "I am aware of the view that having an English name as well as a Welsh name for this institution is felt by some to undermine the use of the term Senedd and also to undermine the use of the Welsh language.
"I would simply observe that the institution currently has a statutory name in Welsh and in English now, neither of which have prevented the growing use of the term Senedd.
"With the best will in the world, I'm not persuaded that the challenge that we have set ourselves of achieving a million Welsh speakers by 2050 will be impacted by the existence of a statutory term in English for this institution in future."
The bill also includes plans for 16 and 17-year-olds and foreign nationals resident in Wales to vote in assembly elections.
Assembly members are expected to have a final vote on the bill in the coming weeks.
We can only wonder just what former First Minister Carwyn Jones meant by that the use of both Welsh Parliament and Senedd to describe Wales’ legislature will be a “transitional stage”.
Well of course the future Lord Carwyn of Pen-y-bont ar Ogwr, wil not be sitting in Cardiff Bay and ome wonders how much he may use Y Senedd when commeting from the Green Benches /
Labour a nd the other Unionist wil resist the sole of Y Senedd because they fear that the Welsh people may identify with it more than the other Parliament in London.
It is 20 years since the "Assembly" first met and im that time we can only look at Scotland with real legislature , whilst our AMs concentrate on how to spend the pocket money Tory and Labour governments in Westminster se fit to allot us.