From The Telegraph:
The weight of all the electrons in motion that make up the internet at any one moment is equivalent to 50 grams, according to Vsauce, a YouTube channel.
It seems to be inspired by recent reports noting that each book loaded onto and Amazon Kindle e-reader makes the device very slightly heavier.
Using Einstein’s E=mc² formula, which states that energy and mass are directly related, Professor John Kubiatowicz, a computer scientist at the University of California, Berkeley, 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.
Here is the full story, which also includes a video.