GA pilot on base leg to BJC airport reported an NMAC with opposite direction traffic 300 ft below them.

Narrative

Flight was VFR from ZZZ to BJC. I contacted ATC out of ZZZ and received flight following to BJC. At approximately 20 nm from BJC I cancelled flight following; and contacted BJC tower. I requested landing with full stop and they responded enter a left downwind for RWY 30L remain at or above 7000. I entered left downwind at assigned altitude and Tower said they would call my base. Tower called my base; cleared me to land; and said base altitude was at my discretion.As I turned into my base turn heading north a plane appeared coming in the opposite direction (south) at approximately 300 below my aircraft. I did pull back on the yoke and climbed a bit for more separation as the plane flew directly under my aircraft.I was able to complete the base to final turn and landed safely. Tower handed me off to ground and I completed the taxi back to the FBO.I followed all instructions in accordance with BJC tower. I was in complete compliance with all FAA regulations before and during the flight.The aircraft passing below me should not have been in that airspace given flight conditions and airport traffic direction.