  We obtain new bounds for the solution of the variance-gamma (VG) Stein
equation that are of the correct form for approximations in terms of the
Wasserstein and Kolmorogorov metrics. These bounds hold for all parameters
values of the four parameter VG class. As an application we obtain explicit
Wasserstein and Kolmogorov distance error bounds in a six moment theorem for VG
approximation of double Wiener-It\^{o} integrals.
