We explore the feasibility and astrophysical consequences of a new long-range U(1) gauge field ("dark electromagnetism") that couples only to dark matter, not to the Standard Model. The dark matter consists of an equal number of positive and negative charges under the new force, but annihilations are suppressed if the dark matter mass is sufficiently high and the dark fine-structure constant $\hat\alpha$ is sufficiently small. The correct relic abundance can be obtained if the dark matter also couples to the conventional weak interactions, and we verify that this is consistent with particle-physics constraints. The primary limit on $\hat\alpha$ comes from the demand that the dark matter be effectively collisionless in galactic dynamics, which implies $\hat\alpha \lesssim 10^{-4}$ for TeV-scale dark matter. These values are easily compatible with constraints from structure formation and primordial nucleosynthesis. We raise the prospect of interesting new plasma effects in dark matter dynamics, which remain to be explored.