Bring broth and potatoes to a boil in a large saucepan. Reduce heat to maintain a simmer, and cook until the potatoes are tender when pierced with a fork, 12 to 15 minutes.
Add kale, and stir to wilt the kale. Cook, stirring occasionally, until the kale is bright green and just tender, about 5 minutes.
Remove from the heat.
Carefully puree in two batches in a blender or with an immersion blender.