The B18A1 motor came in the '89 - '93 Integra's. The '89 - '91 cars are pre OBD or OBD 0 (meaning they have a very simple electronic ignition and ECM/emissions package). The '92-'93 B18A1 (I think they're still a B18A1) are OBD I and have a far more extensive amount of engine harness wiring and better emissions controls. The '96+ Integra motors (and all cars per the US Government) are OBD II and again have more wiring and more advanced ECM's.

If your planning to drop a B18B engine in your DA Integra (which is most likely pre-OBD) your easiest solution is to convert the B18B over to a pre-OBD engine. That means transfering the sensors, injectors and distributor from your old engine over to the new one. The B18B's existing engine wiring harness will (most likely, I don't know all the details here) not plug into your in-car harness and the car will not start.

What year is your car and what year is the engine you will be getting?