Cut through the trunk of the weeping willow tree with a chainsaw as close to the ground as possible. Use proper protection such as thick gloves and protective eyewear when undertaking this activity, as cutting down a tree can be both difficult and dangerous and should only be attempted by those with the proper training to do so.
Remove all remnants of the now-fallen tree. Be certain to police for all twigs and shoots, as even the smallest bit of a weeping willow can begun to run a root system again.
Paint the stump of the tree with a non-selective herbicide. The stump draws the herbicide into the root system, killing the roots as well as the stump. It is suggested to paint the herbicide onto the stump as opposed to spraying, as the non-selective herbicide will kill any plant life with which it comes into contact.
Repaint the stump with herbicide every other day for the next two weeks to assure that the root system is thoroughly terminated.
Dig around the stump of the tree in a roughly 5-foot radius. Dig down until the larger root runners are visible. This will make it easier to actually pull the stump out of the ground.
Remove the stump via the use of a stump-puller. Again, wear the proper protective gear when making use of this, or any other heavy machinery.
Dispose of the now-dead stump in a manner approved by your local and state laws.