There are two answers here. Assume the internet contains about 2000 petabytes of data - that is 2 million terabytes, or 16 billion gigabits.
2 million terabytes would fit onto about 500,000 pounds of solid state devices - assuming about 4 ounces per terabyte. That's data you could actually access. But it's not the answer you want.
If you transmitted the entire internet, how much energy (and thus mass) would be used to encode that data? Assume 16 billion gigabits of data, a 1 watt laser diode, and a 1 gigabit per second fiber line that is perfectly efficient, unbreakable and as long as needed. It would take 16 million kilowatt/hours to transmit the Internet. That's 5.76x10^13 Joules (1.6x10^7 kilowatt/hours) of energy to transmit the Internet through that cable. That works out to about 64 grams of mass for the energy encoding the Internet.
(corrected math)
It would take you 500 years to transmit the Internet. The fiber that would hold the Internet from start to finish would weigh in at 1.55x10^20 pounds (a hundred billion billion pounds). It would wrap around the Earth 1.23x10^14 times.