How do we tell our existing clients that we are renaming our company without confusing or alienating them?

Rather than take an entirely new name, which can cause confusion and disenfranchise people who were loyal to the old "brand," you can update your name by turning it into an acronynm.  Look at AT&T, RCA, MGM and many more that were venerable names that needed to update their old images.  If you can't go this route, you can keep some part of the old name and add to it. That will indicate expansion and growth. 

In the case of a complete relaunch with an entirely new name, you'll need to invest in a complete rebranding campaign with emphasis on assuring clients that their old services will remain.