Let G be the group given by the following presentation:

formula here

The subgroup generated by ab is infinite-cyclic and normal, with quotient the dihedral group of order 6, so G is cyclic-by-finite. The subgroups H = 〈a, c〉 and K = 〈b, c〉 are both dihedral of order 6, and G is isomorphic to the free product of H and K amalgamating L = H∩K. We study K0(kG), the Grothendieck group of isomorphism classes of finitely generated projective kG-modules, and, in particular, the dependence of K0(kG) on the choice of field k. As usual, let ℚ, ℝ and [Copf] stand for the rationals, reals and complex numbers, respectively. We prove the following.