The temperature and magnetic field dependence of the in-plane thermal conductivity κ of the iron-arsenide superconductor Ba(Fe1−xCox)2As2 was measured down to T≃50  mK and up to H=15  T as a function of Co concentration x in the range 0.048≤x≤0.114. At H=0, a negligible residual linear term in κ/T as T→0 at all x shows that the superconducting gap has no nodes in the ab plane anywhere in the phase diagram. However, while the slow H dependence of κ(H) at T→0 in the underdoped regime is consistent with a superconducting gap that is large everywhere on the Fermi surface, the rapid increase in κ(H) observed in the overdoped regime shows that the gap acquires a deep minimum somewhere on the Fermi surface. Outside the antiferromagnetic-orthorhombic phase, the superconducting gap structure has a strongly k-dependent amplitude.