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That sounds to me like you want a Finish->Finish dependency. Task B will not finish until Task A has finished, but can be active before Task A has finished.

Click and hold on the Connection button in the toolbar to get the drop-down menu, then select Finish->Finish. Or drag a dependency line from the lower-right corner of Task A on the Gantt chart to the upper right corner of Task B. Or type it into the dependencies column in the outline, if you have that visible.