Fundamental solutions in a space D′(M) of Roumieu ultradistributions are constructed for convolutors f whose Fourier transform $$\hat f$$ is slowly decreasing. The solutions are of exponential growth if $$\hat f$$ satisfies a stronger condition. These results include a constructive proof of the known existence theorem of Chou. For families of convolutors our method yields solutions which depend continuously on parameters.