NMAC BTWN A DEPARTING MLG AND AN LTT.

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WE DEPARTED WITH MINIMUM SPACING (DUE TO ACFT ON SHORT FINAL) BEHIND A BEECH 1900 TURBOPROP; WHICH HAD BEEN ASSIGNED A DEP HDG OF 330 DEGS OFF RWY 29R AT MSP. OUR ASSIGNED HDG WAS 300 DEGS; A DIVERGING COURSE. UPON INITIAL CONTACT WITH DEP CTL. WE ACKNOWLEDGED VISUAL CONTACT WITH THE PRECEDING B-1900; AND WE WERE CLRED TO 7000 FT MSL ON A 300 DEG HDG; AND HE ADVISED THE B-1900 WOULD BE NEBOUND. AS WE CLBED THROUGH 2500 FT MSL; IT APPEARED THAT THE B-1900 WAS BEGINNING A L TURN TO THE SW; RATHER THAN A R TURN TO THE NE. AS WE REALIZED WE WERE CONVERGING; WE TOLD THE CTLR; AND HE GAVE US A L TURN TO 270 DEGS. I LOST SIGHT OF THE B-1900 AS I BEGAN TO ROLL THE ACFT L; BUT THE FO SAID HAD THE ACFT STILL IN SIGHT; CONVERGING; AND HE TURNED THE TCASII CTL TO TA ONLY TO AVOID AN RA; SINCE WE HAD THE ACFT IN SIGHT. HE DID ASK ME TO INCREASE THE CLB RATE FOR SPACING; SO I ABORTED THE NOISE ABATEMENT PROC AND APPLIED CLB THRUST AND INCREASED PITCH. THE B-1900 PASSED DIRECTLY UNDER US APPROX 200 FT BELOW AS ESTIMATED BY THE FO. UPON RETURN TO MSP 3 1/2 HRS LATER; THE ATC WATCH SUPVR EXPLAINED TO ME A MISUNDERSTANDING BTWN SECTOR CTLRS AS TO THE B- 1900'S INITIAL DEP COURSE. THE B-1900 PLTS WERE SUPPOSED TO HAVE US IN SIGHT AS THEY TURNED L; BUT I SUSPECT THEY MUST HAVE CONFUSED US WITH SOMEONE ELSE; AS NO ONE IN THEIR R MIND WOULD DELIBERATELY CREATE SUCH A COLLISION HAZARD. GIVEN SOMETIMES COMPLICATED DEP PROCS WITH NOISE ABATEMENT MANEUVERS; IT IS COMMON TO DISMISS IFR TFC. YOU HAVE ACKNOWLEDGED VISUAL CONTACT WITH AFTER BEING TOLD THEY WILL COMPLY WITH ATC DIRECTIONS THAT WILL RESULT IN INCREASING SPACING; I.E.; A TURN AWAY FROM YOU. IN THIS CASE; THE ACFT DID NOT DO WHAT OUR CTLR ANTICIPATED; AND LUCKILY; WE WERE STILL OBSERVING THE TFC. OTHERWISE; THIS EVENT MIGHT HAVE HAD DISASTROUS CONSEQUENCES.