Only if you also pass the 20 WPM Morse code examination. A CSCE is only valid for 365 days. I'd heard that Novices and Tech Pluses would be getting an automatic upgrade to General when the new rules go into effect. Also note that there are no automatic upgrades. Below is a summary of some of the upgrading possibilities afforded by the FCC's Report and Order of December 30, 1999.
Grandfather credit is also acceptable, per above.