Tournament chiefs said the decision had been taken after strong advice from Ireland's Department of Agriculture. The Irish government was worried that the 10,000 fans expected to travel to Wales could bring the disease back home.
"It was an easy decision to make," said Six Nations chief executive Roger Pickering. "Our thoughts are with the agricultural communities in the UK and Ireland. Rugby takes second spot in this situation."
The A and under-21 games between the two countries at the weekend were also postponed.
Pickering said there was now some doubt over England's trip to Dublin in the fourth round on March 24.
"We will need to see how the outbreak is contained in the next few days. We will be looking for contingency plans if it cannot go ahead," he said.
The disease affects only livestock but can be carried by humans. Ireland has so far been unaffected by the outbreak which has hit several locations around Britain.
The Irish union had sold its allocation of 4,800 tickets for the game but around 10,000 Irish supporters were expected to travel to Cardiff.
The disease has hit rugby before. In 1967, a foot-and-mouth epidemic swept Britain and the Irish government banned the touring New Zealand All Blacks from travelling to Ireland from Britain.
Tournament organisers said in a statement that the match would be rescheduled in late April or early May, stretching the championship beyond the planned final weekend of action in early April.
They confirmed that Saturday's other games in the third round - England v Scotland and Italy v France - would go ahead as planned.
Welsh team manager David Pickering said Wales were disappointed by the postponement but understood the decision.
"It is very disappointing but we fully appreciate the serious consequences of the outbreak," he said. "Our hearts go out to people in the agricultural sector who have been hit by this."
"We were desperate for the game to go ahead but common sense has to prevail."
Wales coach Graham Henry added: "We are disappointed. But we are a small picture in a big picture. The people in the big picture are making the right decisions."
The decision was announced after British Prime Minister Tony Blair chaired a full ministerial meeting to discuss the crisis.
The Irish Rugby Football Union (IRFU) said it had sought the postponement after a meeting with Agriculture Department chiefs.
"This decision is taken on the recommendation of the Department of Agriculture who strongly advised...that the games should not be played in the light of the foot-and-mouth outbreak in Britain, which could have significant consequences in Ireland should people travel to the games," it said.
Tournament chiefs confirmed in a statement that the other two games would go ahead.
But the statement said: "...the Six Nations Committee is continuing to monitor the situation and will remain in constant contact with the unions involved."
England coach Clive Woodward said he believed the livelihoods of farmers should be placed ahead of rugby.
"The plight of farmers is far greater than any rugby match. They are at the sharp end of it," he said.
Reuters
Britain's foot-and-mouth crisis hit the Six Nations on Tuesday when the game between Wales and Ireland in Cardiff on Saturday was postponed in an effort to help prevent the spread of the disease.
