Radel made the announcement Wednesday at a crowded news conference in Florida.
"I'm not going to sit here and make any excuses for what I've done," he said. "I have let down our country. I've let down our constituents. I've let down my family, including my wife. And even though he doesn't know it, I've let down my 2-year-old son."
He said he takes responsibility for what he did, adding, "I want to be a better man."
At a court hearing earlier Wednesday in Washington, Radel told a judge, "I've hit a bottom where I realize I need help" in acknowledging that he purchased 3.5 grams of cocaine from an undercover police officer.
As part of a plea agreement Radel admitted he agreed to buy the cocaine for $250 in a Washington neighborhood on Oct. 29. After the undercover officer gave Radel the drugs federal agents confronted him, court documents show. Radel agreed to talk with the agents and invited them to his apartment, where he also retrieved a vial of cocaine he had in the home, the documents said.
At his news conference in Florida, Radel said he has been struggling with alcoholism and substance abuse "off and on for years." He said he will enter an in-patient treatment program and said he was confident he could overcome his disease.
His office said the leave of absence takes effect immediately and said the congressman had filed the necessary papers with the House clerk. He gave no indication he was going to resign.
Radel had said in court earlier that he wants to "continue serving this country."
Radel's lawyer, David Schertler, said in court that his client had already entered outpatient treatment in Washington and would also seek treatment in Florida. If Radel successfully completes his year of probation the charge against him will be dismissed and he can apply to have his record expunged.
A DEA official who spoke on the condition of anonymity because he was not authorized to release details of the case in his own name said Radel was identified to authorities as a cocaine buyer by his suspected dealer. The dealer had been arrested previously as part of a separate drug investigation led by a federal task force.
Court documents show that when Radel bought the cocaine on Oct. 29 he met with the undercover officer and an acquaintance with whom he had previously used cocaine. The documents said Radel purchased cocaine on several previous occasions.
Karl Colder, special agent in charge of the DEA's Washington field office, said Radel was given no special treatment in avoiding arrest at the scene. He said authorities do not automatically arrest drug buyers in undercover operations, especially if they are part of a larger investigation, agree to cooperate and don't pose a threat to the public. Radel provided information to investigators at the time of the bust and has continued meeting with them since, Colder said in an interview with The Associated Press.
"Collectively there's always a decision that's made in terms of time of arrest, when we plan to arrest," he said, later adding, "It's not uncommon for us not to make immediate arrests on situations like that."
Radel appears to be the first sitting member of Congress charged with a drug offense since former Rep. Frederick Richmond, D-N.Y., was convicted in 1982 on charges of tax evasion and drug possession.
A spokesman for House Speaker John Boehner, R-Ohio, said the allegations are a matter for the courts.
"Beyond that, this is between Rep. Radel, his family and his constituents," Boehner spokesman Michael Steel said.
The Florida Democratic Party released a statement Wednesday calling Radel's conduct "an embarrassment to his district and to the state of Florida" and saying he "should resign immediately."
Radel was elected in 2012 to represent Florida's 19th Congressional District, which includes the Gulf Coast communities of Fort Myers and Naples. He was a radio host before becoming a congressman. He identifies himself on his Twitter profile as a "Hip Hop conservative" and "lover of #liberty," and his Twitter account has remained active in the past few weeks, including on the day of the bust.
Among the bills he co-sponsored during his freshman term is legislation that would allow a judge to impose a sentence below a mandatory minimum, including for drug offenses.
If Radel had been arrested in Florida with the same amount of cocaine it would be a third-degree felony punishable by five years in prison under state law.
Associated Press writers Jessica Gresko, Alicia A. Caldwell, Eric Tucker and Laurie Kellman in Washington and Curt Anderson in Miami contributed to this report.