From the point of view of someone at a robotics company...
I know places like WPI are starting robotics majors now (and yes I am jealous), but to be totally honest, I've been involved in a lot of hiring my company has done, and I don't know what we'd do with them. What we would probably do during an interview is sit down wit them and ask what they're best at/like most: EE, ME, or software. Then they would be put into the right department. It makes our jobs easier if it's a degree name that matches one of our departments.
There are a few caveats, though. Our company (and many like it) have systems engineers, which is what I am, and which I think the robotics engineering majors might ultimately be suited for, but which tends to be a more experienced position (we don't hire entry-level systems engineers... I had to "do time" in other departments first). These positions might match up well with robotics engineers, but the job is basically technical management: how do the customer requirements translate to technical requirements? How can you make sure what the EEs build fit into the boxes the MEs are building? Did anyone bother asking Software what information they need to write the control code? How/when are we testing this stuff? WHY ISN'T ANYONE DOING ANYTHING? Etc... Basically, you have to have that high-level view of life, the universe, and everything, and be able to coordinate it effectively.
Also, some much smaller companies tend to hire people named "robotics engineer" or "robotics researcher" or "electromechanical engineer" or things like that. I don't know enough about what those jobs do to compare them to a standard ME/EE/SW job, but I know places like Vecna, Sarcos, and others have posted openings like that.
And finally, even though I don't know what we'd do with a Robotics Engineer now, that's not to say we'd never know what to do with them. We definitely need well-rounded people at our company, but right now we're still only putting people into individual departments. I think iRobot is the same way from what I remember talking with them. Personally I think industry should definitely change a bit to accommodate these new awesome engineers, but unfortunately I'm not in charge
Yet!
So in conclusion: this was my long way of saying, it doesn't really matter what your degree is. What I think you should ABSOLUTELY do is find a school that has COOL research. Not just "good" research, COOL research. Find the prof(s) who do robotics and get talking to them. See if they let undergrads help out. That's what is most important to a company like mine, not which degree you have. Get whatever experience you can and then whether you're an EE or RE, you'll have an awesome job waiting for you!