I have had success with Tyvek. It's a breathable, tough, lightweight material that can often be had for free. It's used in the construction business as a vapor barrier. Large sheets can be claimed from dumpsters around houses that are going up. It's very crinkly with a weird metallic sound when you first acquire it. Running it through a couple of cycles in a front loading coin-op washing machine will make it much softer. 

I laid it out under my tent, used a sharpie to get the appropriate outline and then cut it out with shears. Then at the appropriate spots I placed some grommets. It's paper thin, weighs next to nothing and for my 2 man North Face Tadpole tent, it rolls up smaller than a can of soda. Mine has lasted for years. 

 what's the weight like of that stuff?

I cant believe i havnt seen this mentioned before, but as a roofer, i have been using the breathable membrane we use under the tiles on a roof,its like gortex for your roof , its the best thing out there, its light, cuts out the damp on the floor,it also has insulating properties ,which means its warm too ,use one piece as a ground sheet and another piece inside ,its so light i even leave it in my tent when i pack it up, next time you see a roof job going on , ask the roofer if he will sell you a few feet of a roll, otherwise you will have to buy a roll from a builders merchants, and its not cheap!

Me and my mates use blue or black visquene. the plastic DPM sheets builders use for laying house concrete foundations on.

Best advice to get hold of a piece would be to speak nicely to your local builder. measure your area out and carefully, using scissors or a sharp thing called a knife, cut out the required shape.

Most builders will have spare off cuts handy. Recycle is a key word here. Usually a £5ver would do the trick.