Exact calculations are presented for the ground state of linear antiferromagnetic Heisenberg chains with open ends. The wave function and the energy are given for chains with N = 4, 6, and 8, and quantum number S = 1/2. It is shown that in the limit for N - ∞ the ground state is probably nondegenerate and the long-range order tends to zero. The short-range order shows a strongly oscillating character due to end effects persisting over relatively long distances. A comparison with closed rings is made and the connection with the observation of a reduction of the zero-point deviation in the neighbourhood of non-magnetic impurities in more-dimensional systems is pointed out.