Stübner began his career with BSG Motor Halle in 1973 and transferred to SG Dynamo Dresden in 1979, where he gained the nickname "Staubsauger" (Vacuum cleaner), scoring 14 goals in 182 league appearances. Stübner was twice DDR-Oberliga champion with the Dresden club, in 1988/89 and 1989/90, and three-time FDGB Pokal winner. After several seasons, Dynamo reached the semi-finals of the 1988/89 UEFA Cup, where they were beaten by VfB Stuttgart. In total, Stübner played in 29 European games with the club, scoring two goals.
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