If there is no soul, how can there be rebirth? Anatta is often described as "not-self" which I understand to mean that our identities are illusions.  But it's also described as "soullessness" which I think implies that there is no mind other than the brain itself. 

But many Buddhists believe in rebirth.  If there is no soul, how can there be rebirth?
When most people think of "self" they think of some abstract core that is the subject of all experience and the agent of all actions. Buddha taught (and modern cognitive science tends to agree) that upon careful examination there is no such single core. Instead experiences result from interactions of multiple perceptory functions. Similarly, our actions are performed by a conglomerate of functions, without a single agent responsible for all choices.

When the notion of individual rebirth exists as an unexamined belief, it depends on this unexamined notion of "self" as its necessary foundation. Once practitioner is free from illusion of self (in practice, not just conceptually), the notion of individual birth/death/rebirth no longer applies. From this perspective, rebirth is a byproduct of attachment to a (substantial) self.

But because karma at large still continues to function, the infinitely forking/joining threads of causation that once were the subject of attachment continue to participate in activity and serve as causes for subsequent effects. In this sense, rebirth as principle goes on continuously whether we posit an individual or not.

To give this a more practical spin, instead of worrying about rebirth, we practitioners should focus on working towards deconstruction of self by reducing the strength of the illusion through practice of altruism and non-attachment, first and foremost at the microscopic level of thought-to-thought intention and reaction.