The letter, seen by The Telegraph, will heap further pressure on the bishop to quit.

Her comments were also condemned by the senior bishop in Wales, who expressed regret for the "hurt it caused".

Dr Penberthy had posted on the social media platform in March "never, never, never trust a Tory" in response to another Twitter user who had alleged that the Conservatives were planning to abolish the Welsh Parliament or Senedd.

The posting came to wider public attention earlier this month, prompting the bishop to delete her account and issue an apology, admitting that it was not the policy of the Conservative Party to close down the Senedd.

She said she "of course, trust and have trusted Conservatives" and said there were "many honourable people" in the party.

However, one Tory MP branded her "disgracefully divisive" and "not fit to be a bishop".

Simon Hart, the Welsh Secretary, subsequently wrote to the Archbishop of Canterbury on June 8, raising the Government's concerns over Dr Penberthy's politically motivated messaging.

In his response sent on Thursday, Mr Welby, above, told the Cabinet minister he was adding his apology to those already given by the Bishop of Bangor and the Archbishop of York.

He went on: "I... am deeply embarrassed by the use of such language by a church leader. It is absolutely unacceptable, and I am truly sorry.

"I am especially sympathetic because I am very well aware of what it is to be trolled in this way (including by people of all shades of political opinion).

"It is intolerable, and I am determined to work with my sisters and brothers to find better ways to express disagreement."

The Archbishop said reconciliation was a key theme of his ministry, and that meant people from different backgrounds and beliefs need not also agree with each other but needed to learn how to "disagree well".

The Right Reverend Andy John, the Bishop of Bangor, in his reply to the Secretary of State, said: "The Bishop of St David's has apologised for the hurt she has caused as a result of her posts on Twitter, and the Church in Wales has welcomed her statement. I add my apology on behalf of the Church in Wales; I regret the intemperate language that she used and the hurt it caused."

He said he believed the bishop's apology was "sincere" and that she had expressed a desire to improve "the way in which her ministry operates in the public domain".

The Bishop of Bangor said it was clear that there was no distinction for senior clergy between "personal and public statements" and that bishops furthermore needed advice and help in using social media.

Dr Penberthy, a mother-of-four and a grandmother, is the 129th bishop of St David's and was the first female bishop appointed by the Church of Wales. She was elected to the post in 2016.

END