It seems that we have not been keeping the boffin’s busy of late sine they have calculated the weight of the Internet, and that it weights around the same as an average size Strawberry, or to put it better around 50 grams (1.76oz).
Using Einstein’s E=mc² formula, which states that energy & mass are directly related, Prof. John Kubiatowicz, a computer scientist at Berkeley (University of California), calculated that filling a 4GB Kindle to its storage limit would increase its weight by a billionth of a billionth of a gram, or 0.000000000000000001g.
For a full explanation on how the boffin’s have come to this conclusion watch the video below from Vsauce. The mathematics used to come to this weight are rather dated, for example they talk about 75 / 100 million servers, that would have been the case 5 years ago, but since there there has been massive expansion in data centres and the like of Google, Amazon & Microsoft have built many mammoth datacentres.