The council's constitution states that "proceedings at meetings may not be photographed, videoed, sound recorded or transmitted in any way outside the meeting without prior permission of the chair".
Back in July 2011 independent news site Wrexham.com reported that the council had "maintained its ban on the use of Twitter in council meetings".
They said at the time that its reporters had "been told they can't send tweets from council meetings", on a number of occasions.
Today the ban came into the spotlight again when the Daily Post sent along a reporter to cover the Customers, Performance and Resources Scrutiny Committee, prompting editor Alison Gow to take to Twitter to voice her anger at the rules.
She told Journalism.co.uk that while she is "not saying that the council has made a bad decision here", in reference to the specific meeting today, she is "really unhappy" about the general rule banning use of social media.
"We have never come up against this before," she said.
The issue is that there is "no consistency", she added, with other councils even live streaming meetings to the public online.
"I just think saying you can't tweet is ridiculous in this day and age. If you have got people sat in the meeting saying 'this is happening', it makes democratically elected members far more accountable."
She added: "I just do not understand why we would not be able to live cover a council meeting that is discussing the council development plan", when following guidance issued in December 2011, journalists in England and Wales were told they no longer had to seek permission from a judge to report live from court.
But Wrexham council said that under Section 45 of the constitution, the use of social media in council meetings is not allowed.
The specific section states that proceedings cannot be "transmitted in any way outside the meeting" without permission.
"Failure to comply with this Standing Order may invoke Standing Orders 15 and 16 relating to Disorderly Conduct and Disturbance by members of the public."
Wrexham.com reported on the matter again in December last year, after a councillor also spoke out against the rules. At the time Wrexham.com said they "welcome the challenging of the archaic rules that favour other news mediums, and hope more members will now press to get the rules changed".
"It would mean embracing the technology that [the] council are providing its members, and allow them, the media and the public full use.
"To be clear this goes further than Twitter, we would love to ‘liveblog’ meetings or use whatever communication method is current and possible!"
Speaking to Journalism.co.uk today Wrexham.com founder Rob Taylor added: "It's a farcical situation".
Gow has tweeted to say the Daily Post is seeking permission to live tweet this afternoon's council meeting discussing the budget.
Update 1.10pm: Gow has since tweeted that permission to tweet the budget meeting was denied.
Update Thursday 10 January: Wrexham.com has reported that on Wednesday night (9 January) it was granted permission to send "text messages, tweets and emails from a Wrexham Council meeting".
In an addition to our original story, it is also worth noting that in February 2011 former Parliamentary Under Secretary of State in the department of communities and local government Bob Neill wrote to council leaders in England about access to meetings.
While this only applied to councils in England, and not Wales, the letter interestingly makes specific reference to the treatment of those "blogging, tweeting or filming" council meetings, with Neill stating that he wanted "to encourage all councils to take a welcoming approach to those who want to bring local news stories to a wider audience".
Update Friday 11 January: In a statement a Welsh government spokesperson said: "The Minister has placed on record his support for the principle of broadcasting the proceedings of council meetings that are open to the public. The plethora of modern technology channels now readily available to the public means that this is possible without prohibitive expense and should be seen as a means of increasing public engagement with their local government."
They added that "the minister has recently written to the Leader of each County and County Borough Council in Wales asking then what arrangements they have made or plan to make in relation to broadcasting Council meetings. He will be considering their responses shortly."