Their recording of the 1986 single - which has also been a hit for Gun and Spice Girl Mel B - will be released as a download on March 16 with a CD released the following day.
Little Mix star Perrie Edwards said: "We've all seen people push themselves and do incredible things for Sport Relief so it's a dream come true to be releasing the official single this year."
Sport Relief takes place from 21 to 23 March with members of the public invited to raise money by running, swimming or cycling in the Sainsbury's Sport Relief Games.
X Factor winners Little Mix have become the first British girl group to debut in the US top five beating even the Spice Girls.
The foursome shot into the Billboard chart at number four after selling 50,000 copies of their first release DNA.
Little Mix on top as they beat the Spice Girls by reaching the US top-5 with their debut album. Credit: Niall Carson/PA & Doug Peters/EMPICS Entertainment
They even earned praise from the notoriously hard to please X Factor boss Simon Cowell, who sent an online message saying: "Congratulations Little Mix. Very proud of you."
Little Mix, whose hits include chart-topper Wings, responded to Cowell with a message on Twitter which read: "Thank you so much Simon. Thank you for all your support from day one."
The Spice Girls entered the US chart at number six when they released first album Spice in 1997.
