When you take a broken picket from a fence, carefully remove an unbroken picket to serve as a pattern for the replacement. Trace its shape onto a knot-free board that is the same width and thickness as the original.
Cut the new picket with a band, saber, or jigsaw. Sand it, rounding the edges slightly so that water will run off. To seal out moisture, paint the pickets and rails where they will join.
While the paint is still wet, nail the two pickets on with galvanized finishing nails. Set the nailheads below the surface, fill the holes with wood putty, and paint or stain the new picket to match the rest of the fence.