thread testing done flag

I haven't fully grasped all of what I've read about synchronized
access to data from threads, and haven't seen examples doing exactly
what I want to do. Can you tell me if this will work.

I want to have 2 threads, with one thread doing non-blocking I/O, and
the other doing other things. When the later thread is done, it needs
to signal to the thread doing I/O, to wrap it up and quit.

Would something like this be a problem:

class A {
  private boolean done = false;
  class B extends Thread {
    private A caller;
    public B(A c) { caller = c; }
    public void run() {
      while (true) {
        // do stuff
       if (caller.isDone()) {
          // wrap it up
          return;
       }
      }
    }
  }
  public static void main(String[] args) {
    B b = new B();
    b.start();
    // do stuff
    done = true;
    b.join();
  }
  // I think synchronized guarentees that the instance of B
  // doesn't use a local copy of the boolean value
  public synchronized boolean isDone() { return done; }
}