VERY CLOSE NMAC WHEN NORDO SMA CROSSES FINAL APCH COURSE AT MDW AT NIGHT.

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TURNING R FINAL ON VISUAL APCH 13R AT MDW; PIC NOTICED LIGHT CO-ALT; CONSTANT BEARING AT 11 O'CLOCK. PIC INCREASED BANK SMOOTHLY TO BEYOND 30 DEG AS WHITE LNDG LIGHT BEGAN TO TAKE SHAPE ATTACHED TO AN SMA HEADED APPROX OTHER DIRECTION. SMA PASSED CO-ALT; UNDER L WING/BELLY OF MLG. WE WERE IN 40 DEG R BANK. DISTANCE APPROX 150' OR LESS. IF MLG DID NOT SEE AND AVOID THIS OTHER ACFT THERE WOULD HAVE BEEN COLLISION ON CENTERLINE RWY 13R ABOUT 2 1/2 MI OUT. MANEUVER CAPABILITY OF MLG JUST WITHIN LIMITS TO MISS HIM AS IT WAS. DUE TO STORMS APPROX 20 MI W; AND THE LITTLE ILLUMINATION OF SMA'S LIGHT; IT WAS VERY FORTUNATE THAT HE WAS SEEN. ACFT WAS SO CLOSE THAT PIC SAW SILHOUETTE OF ACFT. HIS LIGHTS WERE VERY DIM. THIS SMA WAS NOT TALKING; SQUAWKING; SEEN OR HEARD BY TWR; NOR LOOKING. HE WAS AN ENDANGERMENT TO OVER 60 PEOPLE. IF NOT FOR EXTRAORDINARY LUCK; HE WOULD HAVE NOT BEEN SEEN; AND THEREFORE BEEN HIT. TWR RPTED THAT THIS ACFT HAD A STUDENT PLT; WITH RADIO DIFFICULTY; SOLO; AND AT WRONG ARPT.