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An approximate analytical solution of a laminar jet discharged into an infinitely long cylindrical cavity with a dead end is developed. The employed method assumed velocity profiles whose parameters were determined by solving the integral forms of the continuity, momentum, and energy equations. A novel feature of the method is the use of calculus of variations to obtain a minimum dissipation solution. The solution predicted a few fundamental characteristics of such a jet. First, the jet expands asymptotically to a finite width, which is 52% of the cavity width and is independent of the initial jet radius. Second, after the jet is fully expanded, the centerline velocity decays nearly linearly and reaches stagnation. Thus, the model predicted that in a realistic cavity with a finite length far larger than the theoretical jet penetration, the flow stagnates upstream of the dead end and does not turn around at the closed end. Significantly, the model predicted that with proper scaling, the behavior of the jet for various radii and jet velocities can be collapsed to a single characteristic. The model predictions were compared to CFD simulations and experimental measurements and were found to be in good agreement.