Having just started on pretty much the same thing a few months ago (on a ten-year-old commercial project, originally written with the "C++ is nothing but C with smart structs" philosophy), I would suggest using the same strategy you'd use to eat an elephant: take it one bite at a time. :-)

As much as possible, split it up into stages that can be done with minimal effects on other parts. Building a facade system, as Federico Ramponi suggested, is a good start -- once everything has a C++ facade and is communicating through it, you can change the internals of the modules with fair certainty that they can't affect anything outside them.

We already had a partial C++ interface system in place (due to previous smaller refactoring efforts), so this approach wasn't difficult in our case. Once we had everything communicating as C++ objects (which took a few weeks, working on a completely separate source-code branch and integrating all changes to the main branch as they were approved), it was very seldom that we couldn't compile a totally working version before we left for the day.

The change-over isn't complete yet -- we've paused twice for interim releases (we aim for a point-release every few weeks), but it's well on the way, and no customer has complained about any problems. Our QA people have only found one problem that I recall, too. :-)