The advert says Sellafield is an environmental hazard
And he told the BBC the government's nuclear policy was "fundamentally flawed".
Local and environmental campaigners, which have been fighting a legal battle against the plant, applauded the advert, saying the Irish had "good cause" to fear Sellafield.
But the Department of Trade and Industry said: "All radioactive discharges from UK nuclear sites are stringently regulated to rigorous standards to ensure that health and the environment are properly protected."
Sea pollution
The Mox plant can reprocess plutonium and uranium into a powerful energy source.
But the Irish Government argues that the plant breaks international laws on sea pollution, in addition to its safety and security concerns.
It has already started legal action against the new facility at a United Nations tribunal, and signalled its intention to do so through the European Court of Justice.
The advertisement said: "Sellafield poses an unacceptable and unnecessary risk to our environment.
"Furthermore, in the aftermath of the September 11 assault... we also believe that Sellafield poses a grave security risk to both our countries."
Mr Ahern told the BBC: "We believe having enormous amounts of nuclear waste treading its way up and down the Irish Sea is a fundamentally flawed policy and one we object to strongly."
A British Nuclear Fuels spokesman said it would be inappropriate to comment on the advertisement, saying it was "an issue between the two governments".
But Janine Allis-Smith, a spokeswoman for Cumbrians Opposed to a Radioactive Environment, said: "We fully support this action and can understand how they feel.
"They carry all the risks of Sellafield both in terms of possible terrorist attacks and pollution.
"There are no benefits for the Irish whatsoever. We fully support them in objecting to the operations at Sellafield."
A DTI spokesperson said there was no need for heightened concerns in the wake of the 11 September attacks.
"Prior to September 11, we had in place rigorous security measures at all our nuclear sites.
"In light of the terrorist attacks in the US, these were reviewed and we now have in place measures appropriate to the current level of risk."
Defence
Friends of the Earth, which with Greenpeace next week continues its legal battle to halt the plant, said: "Mox is highly dangerous and uneconomic. It must not be opened."
The two groups will on Tuesday go to the Court of Appeal to challenge the recent High Court decision that the government had acted legally in allowing the plant to go ahead.
"It is time the government turned to the future and invested in clean, green energy for the 21st century. It should abandon the discredited technology of a previous era," the FoE's energy campaigner Roger Higman added.
Prime Minister Tony Blair will put forward a defence of Sellafield when he meets Mr Ahern in Dublin next week.
The plant was built in 1996 and is due for commissioning on 20 December.