Dave Kovic (Kevin Kline) runs a temporary employment agency in Georgetown, Washington, D.C.. As a sideline, he makes appearances impersonating President William Harrison "Bill" Mitchell, whom he has an uncanny resemblance to (and who is also played by Kline). One Mitchell's devious Chief of Staff, Bob Alexander (Frank Langella), hires Dave to make an exit at an appearance of President Mitchell, to cover up Mitchell's extramarital affair with White House staffer Randi (Laura Linney).
When the real President Mitchell suffers a stroke while having sex with Randi, he is left in a coma. Alexander sees an opportunity, and along with Communications Director Alan Reed (Kevin Dunn), he arranges for the President's comatose state to be kept secret. They then con Dave into impersonating the president on an ongoing basis by telling him that the country would suffer if the truth was revealed or if Vice President Gary Nance (Ben Kingsley), who they claim is mentally ill, took office. Apart from them, only his Secret Service bodyguard, Duane Stevenson (Ving Rhames) and the medical staff tending to the real President Mitchell in the White House basement (and who are being paid hush money by Reed), know the truth. Neither Mitchell's mistress Randi nor First Lady Ellen Mitchell (Sigourney Weaver) is informed of the switch.
Alexander's true agenda is to exercise the power of the presidency through Dave as a proxy in the short-term, while in the long term framing Nance for illegal activities, allowing Alexander to succeed Nance as Vice President, and ultimately become President himself. However, Dave gradually sets his own course of action, which revives Mitchell's popularity with the public and mystifies both the Washington media (several well-known personalities make cameo appearances as themselves) and the First Lady, whose public support of her husband has been a cover for bitter marital strife in their private life. Michell calls on his friend Murray, an accountant from Baltimore, to rework the national budget in order to find $650 million. He uses the money to save a favorite program of the First Lady, a homeless project. The First Lady tricks him into exposing his ruse. Caught, he convinces Stevenson to take her to see the real President Mitchell, who is not expected to live. The First Lady has a minor breakdown and decides to leave the White House. Dave realizes he has a chance to actually help the nation and convinces her to remain and keep up the ruse for a while longer. Despite Ellen's cynicism, she is soon charmed by Dave's sincerity.
When Alexander realizes that Dave is no longer his puppet, he threatens to expose him, but Dave stands up to him and points out that Alexander has more to lose from revealing the fraud. Dave then holds a news conference announcing that he is firing Alexander, and proposing a comprehensive full-employment program to Congress.
Dave meets Vice President Nance, who had been sent off on a goodwill tour of Africa. Nancu chastises "Mitchell" for framing him in a financial scandal. Dave realizes that the affable Nance isn't mentally ill as Alexander had insisted, but that Alexander has been using him. Alexander tries to destroy Dave by implicating him in the same financial scandal - only this time, as Communications Director Reed tells Dave, the real President Mitchell is actually guilty. At the same time, Alexander pushes his own candidacy for the presidency.
During the uproar over the scandal, Dave refuses to stop pushing his employment bill. He also bonds with Nance one night when Nance discusses his rise from being an unhappy shoe salesman to being the Vice President. Dave and Ellen agree that Nance is a genuinely public-minded man, and would make a fine president.
In a joint session of Congress, Dave, as Mitchell, admits to his role in the scandal, but introduces evidence (provided by Reed) proving that Alexander was the mastermind and that Nance was innocent. In the middle of making a formal apology to the nation, Dave fakes a stroke and is switched with the still-comatose Mitchell. Mitchell is then declared incapacitated under the 25th Amendment, and eventually dies. The groundswell of affection generated by Dave's brief time in office is transferred to the real Mitchell, as the public and the media mourn his death. The ruse apparently remains undiscovered; although Bob Alexander is later arrested and indicted, there is no indication that he reveals the secret, since doing so would compound his liability. Nance becomes President and promises to carry out "Mitchell's" employment program.
Five months later, Dave is running his own campaign for city council. Former First Lady Ellen Mitchell, now a widow, comes into the office "seeking a job," and they kiss. When he closes the shades in the window of his office door, Stevenson steps in front, wearing a campaign badge for Dave.