  We show that the Grothendieck period conjecture holds for the Kummer surface
associated with the square of a CM elliptic curve. This means that the period
isomorphism is dense in the torsor of motivic periods. In other words, the
isomorphism is dense in the torsor of motivated periods, and motivated classes
on powers of the surface are algebraic. The point is that the motive has a
non-trivial transcendental part, but belongs to the Tannakian category
generated by the motive of a CM elliptic curve.
