Mr Amari told a press conference in Tokyo on Thursday that he is resigning.
He was under pressure following allegations of corruption made by the Japanese magazine Shukan Bunshun.
Shukan Bunshun is one of a number of weekly magazines which specialise in scandal. Most respectable people say they do not believe all things they read in them but a good story will be picked up by the quality newspapers and the Japanese broadcaster NHK.
In this case, the allegations, which were published last week, were that Mr Amari and his aides were given money and gifts worth twelve million yen by a construction company in return for favours linked to land ownership.
Mr Amari’s denial was less than robust. He said he did receive money but denied it was a bribe.
The international press concluded Mr Amari’s departure is a blow for Prime Minister Shinzo Abe, although some Japanese journalists wonder if it was part of a power struggle.
“This is possibly the biggest scandal the Abe administration has faced,” said the BBC’s Mariko Oi.
It turned out that Mr Amari, who said he was trying to fight deflation, resigned just a day before the Bank of Japan picked up another weapon in that ongoing battle.
On Friday it announced it would introduce negative interest rates for the first time.
That move is designed to persuade the commercial banks to lend more money to businesses and to encourage Japanese people to spend some of their savings.
This policy will run alongside the Bank of Japan’s existing programme of buying Japanese government bonds.
The problem is that as it does so, it amasses more public debt.
Professor Masanaga Kumakura from Komazawa University complained about this in an article, published in East Asia Forum, saying that Japan is sinking “in a sea of money”.
The Bank has already bought more than 300 trillion yen (about US$ 2.5 trillion) of bonds and has pledged to go on buying more until the rate of inflation rises to two per cent.
Professor Kamakura said that this misguided policy is based on an inaccurate view of deflation.
Elsewhere, in an article for Forbes, Bryan Rich suggested that the Bank of Japan now holds the steering wheel of the global economy.
He claimed its policies will be beneficial to investors because it wants Japanese stock prices to rise and it is also aiming for a weaker yen.
But like Professor Kamakura, he is concerned that Japan will run up more debt without a credible plan to repay it.