I can see it two ways - if all the $CO_2$ has turned into bubbles I would imagine it is lighter. However dissolved $CO_2$ atoms are probably heaver than the $H_2O$ atoms they replace, or fit in between.
As a general rule, the volume of dissolved solutes in water is less than the volume an equal mass of the solutes will occupy as a solid, this is just because water molecules fill up space so badly, nearly anything can partially fit in interstitials. Since dry ice is already 50% heavier than water, CO2 solution is heavier than water, by unit liquid volume.
But as you say, there is an out-of-equilibrium outgassing process--- you produce gas which is much lighter than the water or the CO2. If you weigh soda water which is out of equilibrium with an equal apparent volume of water (fill up the glass to the same height, ignoring the volume lost to bubbles), the water can be heavier than the soda-water, just because there is more liquid volume in the water.
To see that there is a limit where this is true, consider filling up a beer glass 50% with Guiness, and compare with a full beer glass of water. The same holds for very fizzy drinks that can sustain a good volume of bubbles. For ordinary soda, the volume of bubbles trapped inside depend on the surface property of the glass, and the answer depends sensitively on whether the total gas volume contained in bubbles inside the soda is at a certain time greater than the total extra mass of dissolved CO2 in the bulk liquid. This depends on the precise configuration in a complicated way.