We prove a result in the theory of zero-sums. Let x1,...,xr be a sequence of elements of Zn, the integers modulo n. How large must r be to guarantee the existence of a subsequence xi1,...,xin and units a1,...,an with a1xi1+...+anxin=0 ? Our main aim is to show that r=n+b is large enough, where b is the sum of the exponents of primes in the prime factorisation of n. This result, which is best possible, could be viewed as a units version of the Erdos-Ginzberg-Ziv theorem. This result was conjectured by Adhikari, Chen, Friedlander, Konyagin and Pappalardi in a recent paper. In that paper they also proved a related result concerning sums weighted by signs -1 and 1. We give a new proof of their result. We also discuss some questions related to our results.