He said Ms Osthoff, 43, was at the German embassy in Baghdad and that her drver who was kidnapped with her was also expected to be released.
“The kidnappers have announced that they would also release her driver. We are very happy about this outcome,” the minister said.
“After all the weeks of waiting and uncertainty we share the joy of her family and friends whole-heartedly.”
The archaeologist’s brother Robert Osthoff earlier told German n-tv television station: “My sister is free”.
Mr Osthoff said that ironically his mother and sister Anja, who had launched an emotional plea for the hostage’s release, “are not available today” and had therefore not yet been told the news.
Converted to Islam
Susanne Osthoff, 43, a convert to Islam who has lived in Iraq for 10 years, was seized with her driver on November 25 in the Nineveh region of northwest Iraq.
Mr Steinmeier expressed his gratitude to those who had worked for her release, including the German embassy in Baghdad and a crisis team in the foreign ministry, and thanked the public for their solidarity with the hostage.
Ms Osthoff hails from the southern German state of Bavaria and the mayor of her hometown of Glonn, Martin Esterl, said that the inhabitants were overjoyed to hear of her release.
“It is a great relief for all of us. We will be delighted to welcome her back in person, but this will of course take some time because first she will need a lot of rest.”
The president of Germany’s Central Council of Muslims, Nadeem Elyas, also welcomed Osthoff’s safe release.
“May God protect our country, the German people and all of the world from such criminal acts in future,” he said.
Ms Osthoff was the first German national to be kidnapped in Iraq and her plight posed the first serious crisis for Chancellor Angela Merkel, who took office in late November.
Mrs Merkel had appealed to the kidnappers to release her, but said Berlin would not be blackmailed.
Her predecessor, former chancellor Gerhard Schroeder, also urged the kidnappers to free her, pointing out that Ms Osthoff had worked hard to help Iraqis.
“I appeal to you to act humanely and show mercy. She has made Iraq her home,” he said.
Mr Schroeder’s government refused to send troops to Iraq, but Germany has been helping to train Iraqi policemen, though not on German soil.
According to German press reports, the kidnappers set an ultimatum of a few days for Germany to stop training Iraqi police officers as the price for Ms Osthoff’s release shortly after she was seized.
Since she disappeared, however, the only word from the kidnappers had been a video released to a German TV station purportedly showing her and her driver blindfolded and surrounded by armed men.
As the hostage saga continued, public support in Germany for Ms Osthoff’s plight grew.
About 300 Germans had held a candle light vigil for the hostage in central Berlin on Wednesday following a public appeal for support from her sister, who told national television: “Do not forget her, support her”.
The plea came as ZDF public television reported that the German government managed to establish contact with “intermediaries” who would negotiate with Ms Osthoff’s kidnappers.
There was no official confirmation of the report, but the German weekly magazine Focus reported in an advance extract of its Monday edition that the government believed she was still alive.
Citing sources in the foreign ministry crisis unit, Focus said she was seized by a group calling itself the Army of the Mujahedin, which initially thought she was a spy working for a Western government, but later realised she was not.
Ms Osthoff was married to an Arab and speaks fluent Arabic, according to her family.
They said repeatedly that she had tried to help Iraqis in need and also to preserve their country’s heritage.
“My sister has a very big heart. She did not want to get involved politically. She only wanted to help,” Robert Osthoff said.