Abstract <p> We consider a two-dimensional hypersingular integral equation in a convex boundeddomain whose boundary is a smooth curve. The equation contains an integral operator withintegral understood in the sense of Hadamard finite part. We study the existence of solutionshaving a power-law singularity on the boundary of the domain: the solution is sought in the classof functions represented as the ratio of a smooth function to the root of the distance from a pointto the boundary. We prove that the action of an integral operator with a power-law polesingularity of the third order on a function in the class in which the solution is sought givesa function that is Hölder continuous on the entire domain. Further, we prove that thehypersingular equation has a solution with the above-mentioned power-law singularity on theboundary of the domain and indicate a boundary condition under which such a solution is unique.</p>