Define "fizzy." If "fizzy" means "containing dissolved gas," then the density of the solution won't be much different than that of pure water (depending on what gas and how much). If on the other hand we mean "water full of gas bubbles," then of course the density is lower than that of pure water--in fact, eruptions of gas in the Bermuda Triangle have been blamed for the sinking of ships because of this. If half of the volume of a "fizzy water" is gas bubbles, then its density is roughly half that of plain water; a ship floating in the stuff will sink until it displaces twice the usual volume of liquid it needs to keep it afloat--and if there isn't enough ship left to do this, it is lost.
In Carbonated soft drinks carbon dioxide (clue in the name) is used to give the "fizz". Somthing like Cola is carbonated to 4 "volumes" ie 1L of product contains 4L of gas at normal pressure. Now each volume of gas is 0.2% by weight of the product, so in a 1L bottle of cola we would have 1000ml x 0.2% x 4 = 8g of CO2. So assuming no increase in volume (because the gas is disloved) and pure water (density 0.9982)then it would weigh 1006.2g per litre.
Related Topic: Diet Carbonated Drinks are lighter than Regular Carbonated versions, due to not having sugar. I was once surprised at a cookout that aluminum cans of diet soda floated while the regular ones sank. They had been dumped in a tub of cold water chilled with pieces of dry ice [i.e. chunks of frozen carbon dioxide].
@First Anonymous:I think that your assumption that there is no increase in volume leaves the question unanswered. The question could be reformulated as: when a gas is dissolved in a liquid, does the volume increase?
Shake a bottle of fizzy drink. (Make it a glass bottle, to avoind a lengthy discussion of whether it expands when shaken.) Open the cap. Gas hisses out. Consequently, the bottle must now weigh less than it used to.