The problem of asymptotic completeness (AC) is the question whether all pure states can be interpreted in terms of scattering states of particles. The authors study here the two-body AC in massive Euclidean lattice quantum field theories. Schwinger \(n\)-point functions \(S_n(x_1,\dots, x_n)\), \(x_i\in \mathbb{Z}^{d+1}\) with \(d\geq 1\), and Bethe-Salpeter kernel \(K(k,p,q)\) are defined. Let \({\mathfrak H}_{bs}\) be the space containing one-particle bound states, and \({\mathfrak H}^{2(m+\delta_0)}\) the space of the states having energy less than \(2(m+\delta_0)\). They propose the following four hypotheses: H1: Existence of ``one particle states'' (i.e. \(\widehat{S}_2(p_0,{\mathfrak p}))\). H2: \(K\) is analytic in \(|\operatorname {Im} p_i| 0\) \((i=0,1,\dots, d)\) and \(\varepsilon>0\). H3: \(K= \eta 1+\eta^2 K_1(k,p,q)\) for \(\eta\geq 0\), and \(K_1\) satisfies H2. H4: \({\mathfrak H}^{2(m+\delta_0)}\) basically consists of ``two-particle states''. The following theorem is proved: Theorem. \({\mathfrak H}^{2(m+\delta_0)}= ({\mathfrak H}_{\text{in,out}} \oplus {\mathfrak H}_{bs})\cap{\mathfrak H}^{2(m+\delta_0)}\) holds under H1, B2, and H4. If in addition H3 holds, \({\mathfrak H}^{2(m+\delta_0)}= {\mathfrak H}_{\text{in,out}}\cap {\mathfrak H}^{2(m+\delta_0)}\). The condition \(d=1\) makes the proof constructive.