Q: Doesn't atheism inevitably lead to nihilism?
A: No. Atheism doesn't lead inevitably to anyplace, any more than starting in St. Louis inevitably means you have to end up in New York. It's a starting place, not an entire route; the final destinations at the end of the trip are as many and varied as there are atheists.
Regarding nihilism specifically -- the general philosophical position that our existence (action, suffering, willing, feeling) has no meaning -- it's certainly possible that someone starting out with "I don't believe in god" might end up there.
But that's just one possibility among uncountably many other ones.
For instance, an atheist might believe that while there is no God, there are other planes of existence to which we might be able to aspire, and thus continue our lives as trans-humans. Or they might believe that we humans are capable of creating our own meaning, which gives our lives purpose and direction. Or that there certain actions are more aligned with the natural order of the universe, and that acting in accordance with those rules is healthier. Or that they live on in their children, and this is the meaning for which they live.
You get the idea -- there are lots of different ending positions you might arrive at after beginning with a lack of god-belief. To argue otherwise would be like someone telling a theist "Ah, since you believe in a god, then your fondest wish must inevitably be to die so you can go join Him, and therefore this life has no real meaning for you."
In general, whenever you hear someone say "Atheism means ...", swap out "atheism" for "theism" and see how much sense it makes. Atheism and theism are both very, very broad terms; trying to tie either of them down to very specific positions generally leads to errors.
(Update: I rarely update this site any longer. For more discussion on this topic, please see Jason Rosenhouse's excellent post "Must Atheists Be Nihilists".)