This is NOT TRUE.
NASA, Snopes, Scientific American and many other sources on the internet refuted the claim. Here is a summary of the story, taken mainly from Scientific American:
At first both the USSR space program and NASA used pencils for their missions:
Originally, NASA astronauts, like the Soviet cosmonauts, used pencils, according to NASA historians. In fact, NASA ordered 34 mechanical pencils from Houston's Tycam Engineering Manufacturing, Inc., in 1965. They paid $4,382.50 or $128.89 per pencil.
However, pencils are not really good for a space mission, as they are highly flammable in a rich oxygen environment, and can break off the tip which can hurt both astronauts and the equipment:
Pencils may not have been the best choice anyway. The tips flaked and broke off, drifting in microgravity where they could potentially harm an astronaut or equipment. And pencils are flammable--a quality NASA wanted to avoid in onboard objects after the Apollo 1 fire.
The Fisher Pen Company developed a space pen by themselves at an investment of 1 million dollars, and only after development came to NASA to propose the pen, and NASA tested it for 2 years to make sure it stands up to their standards:
Paul C. Fisher and his company, the Fisher Pen Company, reportedly invested $1 million to create what is now commonly known as the space pen. None of this investment money came from NASA's coffers--the agency only became involved after the pen was dreamed into existence. In 1965 Fisher patented a pen that could write upside-down, in frigid or roasting conditions ...
That same year, Fisher offered the AG-7 "Anti-Gravity" Space Pen to NASA. Because of the earlier mechanical pencil fiasco, NASA was hesitant. But, after testing the space pen intensively, the agency decided to use it on spaceflights beginning in 1967.
Since then and until this day, both Russian and American space crew have been using the Fisher company antigravity pen, and it's even offered for sale for the general public:
According to an Associated Press report from February 1968, NASA ordered 400 of Fisher's antigravity ballpoint pens for the Apollo program. A year later, the Soviet Union ordered 100 pens and 1,000 ink cartridges to use on their Soyuz space missions, said the United Press International. The AP later noted that both NASA and the Soviet space agency received the same 40 percent discount for buying their pens in bulk. They both paid $2.39 per pen instead of $3.98.
Since the late 1960s American astronauts and Russian cosmonauts have used Fisher's pens. In fact, Fisher has created a whole line of space pens. A newer pen, called the Shuttle Pen, was used on NASA's space shuttles and on the Russian space station, Mir.