The current plan is to begin offering replacements on Monday, January 20th. As we noted in November of last year, this initiative has been in the works for several months as Apple needs to source individual screens for replacement and supply each of its stores within special calibration machine.
The repair program would let an Apple retail store fix a user's display in about an hour without the need to replace the entire device. This may also be easier for customers as they won't need to reinstall any applications or data onto a replacement device. If your broken screen isn't covered under warranty, the repair will likely cost $149.
A few select Apple stores have already started conducting the iPhone 5c screen replacements but Monday is said to be the full-scale rollout. Apple is also working towards introducing in-store screen replacements for the iPhone 5s; however, the company has yet to roll that out on a large scale.