Great Western Bancorp, which has been family owned for 35 years, is based in Sioux Falls, South Dakota.
"It is a strong, well-run bank with a robust history of expansion, a strong deposit base and sound financial performance," Stewart said.
Great Western Bank was also active in the commercial lending, wealth management distribution and insurance agency sectors.
The deal is subject to regulatory approvals in the U.S. and Australia.
Australia's major banks have benefited from 16 straight years of economic growth and have escaped exposure to the troubled U.S. subprime market, which has led to billions of dollars of writedowns elsewhere.
Stewart said the acquisition would have minimal impact on NAB's financial situation and the positive group outlook was unchanged.
The takeover was expected to be earnings per share neutral by year three, about 0.5 percent dilutive in the first year, and have no impact on dividends.
NAB, which also owns the UK's Yorkshire Bank and Clydesdale Bank, earlier this month posted a 20 percent increase in second-half profit.
The bank recently completed a three-year restructuring program, designed to reduce costs and take it forward after a series of stumbles, including losses form unauthorized foreign currency trading.