The purpose of this note is to obtain a restriction on the fundamental groups of non-elliptic compact complex surfaces of class VII in Kodaira's classification [9]. We recall that these are the compact complex surfaces with first Betti number one and no non-constant meromorphic functions. This seems to be the class of compact complex surfaces whose structure is least understood. The first and simplest examples are the general Hopf surfaces [9, III]. Then there are various classes of examples found by Inoue [5, 6], and which have been studied in more detail in [11]. The only known topological restriction beyond the first Betti number is that intersection form in two-dimensional homology is negative definite. There seems to be little known as to how wide this class of surfaces is. We prove the following theorem.