Known for his cautious steering of interest rate policy as head of the powerful Fed, Mr Greenspan proposed to Ms Mitchell on Christmas Day after the two had dated for more than a decade.
"We have a huge amount in common. It's a really great match," he told the Washington Post newspaper. Mr Greenspan and Ms Mitchell, 50, were both married previously.
Ms Mitchell said that the proposal, which came after the two had spent Christmas morning with friends, was a complete surprise. "We came home and he said, `Do you want a big wedding or a small wedding?'" she recounted. No date has yet been set for the nuptials.
Mr Greenspan, a balding American with big round glasses, has applied the same kind of measured caution to his matrimonial decision as he does to US fiscal policy. He has been dating Ms Mitchell for 12 years.
Asked what effect the news of their engagement might have on Wall Street, Ms Mitchell replied: "I hope they will not become irrationally exuberant."
Mr Greenspan rocked world financial markets earlier this month when he said that the central bank had to be wary when "irrational exuberance" infected Wall Street stock prices. "How do we know when irrational exuberance has unduly escalated asset values?" he said.