Is it possible to become a professor in a field where you don't have a PhD degree?
Yes, it is possible to cross over between different fields following the PhD-level studies. However, in general, this tends to be more applicable to "interdisciplinary" fields that can fall into multiple disciplines. For instance, the engineering department I studied at hired people with PhD's in applied mathematics and physics, because their research fields—in fluid mechanics and interfacial science, respectively—meshed well with the research interests of the department.

To give a counterexample, however, it will be much harder to make the case for a high-energy physics person to move into another discipline, just because it is so strongly identified with physics.

Thus, if you have a PhD in a field such as mathematical economics or econometrics, it will be a lot easier to make the lateral shift. However, if you're in a more "traditional" subfield, that move becomes much more challenging.