Experiments on buoyancy-driven double-diffusive convection sustained by imposed vertical concentration gradients (one stabilizing, the other destabilizing) have been conducted in a thin (Hele--Shaw) isothermal rectangular cell. Novel gel-filled membranes were used to sustain the concentrations at the boundaries. When the destabilizing solute diffuses more rapidly than the stabilizing one, the primary instability leads to traveling waves with a high reflection coefficient at the ends of the cell. The measured critical Rayleigh numbers and frequencies are in reasonable accord with a stability analysis that includes corrections for the finite thickness of the cell and cross-diffusion effects. The weakly nonlinear waves that appear at onset do not stabilize, even very close to the transition, but continue to evolve, eventually becoming a packet of large amplitude plumes. The packet travels back and forth along the cell in a nearly periodic manner. This behavior and the absence of measurable hysteresis are consistent with the present weakly nonlinear analysis which predicts tricritical scaling ([similar to][epsilon][sup 1/4] rather than the usual [epsilon][sup 1/2]) [ital all] [ital along] [ital the] [ital instability] [ital boundary]. However, the range of this scaling in [epsilon] was found to be less than 0.005, which is inaccessible in the present experiments.