We consider the Geometric Connected Facility Location Problem (GCFLP): given a set of clients C ⊂ ℝ d , one wants to select a set of locations F ⊂ ℝ d where to open facilities, each at a fixed cost f≥0. For each client j ∈ C , one has to choose to either connect it to an open facility ϕ(j)∈F paying the Euclidean distance between j and ϕ(j), or pay a given penalty cost π(j). The facilities must also be connected by a tree T, whose cost is Mℓ(T), where M≥1 and ℓ(T) is the total Euclidean length of edges in T. The multiplication by M reflects the fact that interconnecting two facilities is typically more expensive than connecting a client to a facility. The objective is to find a solution with minimum cost. In this paper, we present a Polynomial-Time Approximation Scheme (PTAS) for the two-dimensional GCFLP. Our algorithm also leads to a PTAS for the two-dimensional Geometric Connected k-medians, when f=0, but only k facilities may be opened.