The security guard, identified as Stephen Tyrone Johns, and the gunman were both taken to George Washington University Hospital where Johns died, several media outlets reported.
The gunman armed with a rifle walked into the museum, about two blocks from the Washington Monument, and opened fire on Johns before being shot by two other security guards returning fire, law enforcement officials said. The confrontation occurred just inside the main entrance to the museum.
Police confirmed the name of the suspect as James W. Von Brunn, The Washington Post reported. Officials said the man -- if he is the James W. Von Brunn they suspect -- has connections to U.S. hate and anti-government groups, several media reported.
CNN reported the alleged shooter served six years in prison for the attempted armed abduction of Federal Reserve members in Washington in 1981. He was 62 years old at the time and lived in New Hampshire, the network said.
Washington Mayor Adrian Fenty said the suspect was in critical condition.
Washington Metro Police Chief Cathy Lanier said she didn't know of any statement being made by the suspect, whose identity she said she could not confirm.
"It appears to be a lone gunman" who walked through the museum's entrance with his weapon visible, fired upon Johns and two other security guards returned fire, she said.
"There was no prior information, no prior threats," Lanier said.
The FBI said it had no prior knowledge of any act or threat against the museum.
"At this time, we have no additional intelligence information to indicate threats to area landmarks but will monitor the situation closely," said Joseph Persichini, assistant director in charge of the FBI's Washington field office. "Preliminary indication is that this incident involved a lone suspect."
Lanier said bomb-sniffing dogs swept the nearby area surrounding the museum "as a matter of routine."
"There are no words to express our grief and shock over today's events at the museum, which took the life of officer Stephen Tyrone Johns," the facility posted on its Web site. "Officer Johns, who died heroically in the line of duty, served on the museum's security staff for six years. Our thoughts and prayers go out to officer Johns's family."
The message said the Holocaust Museum would be closed Thursday and its flags flown at half-staff to honor Johns.
One witness, a 19-year-old girl named Maria, told CNN she saw a security guard "down and bleeding on the floor."
Museum visitors told CNN said they heard "booming sounds" before being told to evacuate. Some told reporters they thought the sounds were of something falling or part of an exhibit.
President Barack Obama said he was "shocked and saddened" about the shooting at the museum.
"This outrageous act reminds us that we must remain vigilant against anti-Semitism and prejudice in all its forms," he said in a statement. "No American institution is more important to this effort than the Holocaust Museum, and no act of violence will diminish our determination to honor those who were lost by building a more peaceful and tolerant world."