The only way you could conceivably do this is with a proxy program that's setuid root (i.e., chown root my-proxy-process; chmod 47nn my-proxy-process. And that setuid proxy program takes care of handling security considerations, setuid'ing to the named user, etc.
However the security problems with this should be fairly clear. One way it could be limited is to ensure your non-priveliged process runs with a user in a name group. Then chown thet proxy command with root:myprivategroup and chown it as 4710 so only users that are members of myprivategroup can execute it.
This is probably as secure as it can be. The main thing is making sure the proxy process is good and secure and locked down so only pertinent users can run it through group membership.
If there's any other restrictions you know can be applied to user-supplied processes, then the proxy program could validate that the process confers to those rules. Note that whilst this may be a stop gap, it wouldn't be beyond the realms of technology for someone to craft a "bad" program that passes the tests.
For added security (recommended), use a sandbox or chroot jail as mentioned by mark4o.