Cure will become Northampton's sixth permanent manager since May 2016, after the club dismissed Dean Austin after Saturday's 4-0 defeat at Mansfield.
"In the position we are in we think we needed a manager with a lot of experience of League 2 and Keith has that," Chairman Kelvin Thomas told the club's website.
"He has managed around 400 games at this level and we think he is exactly who we need right now. He took over at Carlisle United in similar circumstances and did an excellent job, not only steering them out of trouble but developing a squad that was able to challenge for promotion in the following seasons.
"He is respected for the job he did at Carlisle United and we believe he will give us strong leadership and he won't be shy to ruffle feathers to get results.
"We were keen to move quickly and after discussions and finalising paperwork today, Keith and Colin have met the players and staff. They were determined to take charge of the team for the game on Tuesday night to try and have an impact rather than just have a watching brief.
October 1, 2018
"We have the talent within the squad at the club and we need someone who can help them turn in the performances required to secure results. As always, it’s going to be a tough job, but Keith and Colin are up for the challenge and want to hit the ground running.
“As part of the incoming appointments Andy Todd has left the club and we thank him for all his hard work and wish him well for the future.”