Mr Abdullah said he had spoken to Mr Karzai by phone, in what is said to be their first confirmed contact since the first round in August.
"We are completely ready for the second round," the former foreign minister said, urging officials to organise a "free, fair and credible" election.
Speaking on the BBC's Newsnight programme on Tuesday, Mr Abdullah said a coalition government was unlikely, but if elections proved impossible for "practical reasons" the two rivals needed to talk to find an alternative solution.
There are concerns that holding a second round of voting in November could lead to a repeat of August's massive fraud, as well as logistical problems caused by winter weather, which could leave much of the north of the country inaccessible.
Lyse Doucet spoke to Kai Eide, the UN special representative to Afghanistan
The UN special envoy to Afghanistan, Kai Eide, said it was encouraging that action was being taken against officials responsible for election fraud, but he sought to downplay expectations about the second round.
"If we also reduce the number of polling centres and remove some of those where we know that fraud took place then I think we are in a bit better situation than we were, " he told the BBC.
"But let us not be under any illusions, we cannot make any dramatic changes in the course of two weeks."
Also, AFP news agency quoted Swedish Foreign Minister Carl Bildt as saying the European Union would not have enough time to mobilise a large contingent of observers for the run-off.
EU monitors observing the August elections said that up to a quarter of votes counted showed indications of fraud.
'All necessary measures'
Our correspondent says it will be a massive task to get so many new officials in place in time for the run-off.
But the UN mission in Kabul says the business of removing some 200 officials and recruiting new personnel had already begun before Tuesday's announcement of a second round.
Speaking in New York on Tuesday, Mr Ban said the UN had learned "quite a painful lesson" after seeing the widespread fraud in August.
KARZAI V ABDULLAH
First popularly elected president of Afghanistan
Opposed Soviet occupation in 1980s
Critics say he has done little to rein in corruption
Tajik-Pashtun, doctor by profession
Senior Northern Alliance leader during Taliban rule
"We realised that it was quite difficult for a young democracy to stand on its own, even with strong international assistance, particularly by the UN," he told the BBC.
"We will also try to visit all the polling stations to make sure that no such fraud can happen," he added.
The secretary general dismissed charges that the UN tried to cover up the extent of the widespread fraud in the first round vote, saying the issue had been not to hide it, but how best to deal with it.
"We have made it clear to the Afghan government, and we made it clear to the Security Council, that there was fraud we wanted to rectify," he said.
"We really wanted to provide a full opportunity to all the Afghan people so they could cast their free vote to elect their next leader."
Since the disputed first round of polling, there has been intensive Western lobbying of Afghanistan's leaders to resolve the weeks of political paralysis.
The White House - debating a request for 40,000 more US troops to be sent to Afghanistan - warned at the weekend that no more soldiers would be deployed until a political resolution was reached.
President Barack Obama welcomed news of the run-off, saying: "It is now vital that all elements of Afghan society continue to come together to advance democracy, peace and justice."
British Prime Minister Gordon Brown also welcomed the "statesmanlike" move.
Initial election results suggested Mr Karzai, the incumbent, had received 55% of the vote, and former foreign minister Mr Abdullah 28%.
But on Monday the UN-backed Electoral Complaints Commission (ECC) deducted hundreds of thousands of votes from the main candidates.
Its investigation focused on 600 of the most serious complaints, and a sample audit of suspect votes at 3,377 polling stations. At 210 polling stations all the ballots were invalidated.
This meant Mr Karzai's total was reduced to below the threshold required for outright victory - 50% plus one vote - indicating a second round was needed.
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