Well, we need to go kill an Ice Wraith, but not just any Ice Wraith, as it has to be one on Serpentstone Isle. Now, Serpentstone Isle is tricky to get to and without knowing where it is you've probably never been there before (unless of course you used this guide to find it, as it DOES house the STANDING STONE named "The Serpent Stone"). Basically, you want to head to the major city of Winterhold and head east, crossing the ice floes until you reach the island. Once your on it, head to the top where the standing stone will greet you, along with an Ice Wraith. Slay the beast and you'll be done, and your next objective will be to report back to Galmar.
Galmar will be surprised you're back (the punk actually bet against us), but he'll let you join after taking an oath to always be loyal to Ulfric Stormcloak (even AFTER you die, which is hardcore). This mission will come to an end now as you're official part of the Stormcloaks (they even give you a set of their threads to prove it). You'll also start the next mission: "The Jagged Crown".
Note: If you had the quest "Join the Imperial Legion" it will now have FAILED, since you can only pick one side or the other.