Richardson had been serving as caretaker boss at the Crown Ground since Paul Cook's decision to join Chesterfield last week and led Stanley to a 1-0 weekend win at Bristol Rovers, ending a four-game losing run.
The 32-year-old former Stanley defender fulfilled a similar role when John Coleman departed to Rochdale last season but takes the reins on a full-time basis from Saturday's FA Cup first-round clash at AFC Fylde.
"It's a bit surreal to be honest," said Richardson in a club statement.
"The conversations that we've had this week have gone really well and I'd like to thank the chairman Peter Marsden and Rob Heys, the board, the players, all of the staff and of course the fans for giving me this opportunity.
"This is fantastic and I'd like to think I could repay everyone with hard work and get some success at the club."
Managing director Heys made it clear Richardson was Accrington's number one choice in the wake of Cook's departure and the appointment comes despite speculation that he might have resumed duties as his predecessor's number two at the Proact Stadium.
"As soon as Paul made the decision to leave Leam was always our first choice to make the step up and take over as the manager," Heys said.
"Over the last eight months as Paul's assistant he has demonstrated all the qualities that we were looking for and I have no doubt that he will do a great job in his new role."
Richardson was part of the Accrington team that won promotion to the Football League in 2006 and went on to play almost 150 times for the club.
- Sports & Recreation