The author introduces a new metric in the space of positive measures on a separable metric space, the convergence in which is equivalent to weak convergence. Let S be a totally bounded metric space, let M be the space of signed measures on the Borel \(\sigma\)-algebra of S, \(M^+\) be the subset of positive measures and let U be the normed space of bounded uniformly continuous real functions on S with sup-norm. A countable set \(H=(\Phi_ n)_ n\) which is everywhere dense in U defines the following metric on M: \[ d_ H(\mu,\nu)=\sum^{\infty}_{n=1}(1/2^ n\| \Phi_ n\|)| \int_{S}\Phi_ nd\mu -\int_{S}\Phi_ nd\nu |,\quad \mu,\nu \in M. \] The author shows that if \(\mu_ n\in M\) and \(\mu_ n\to \mu\) weakly, then \(d_ H(\mu_ n,\mu)\to 0\). If \(\mu_ n\in M^+\), then the converse statement is also true, i.e. \(d_ H(\mu_ n,\mu)\to 0\) implies that \(\mu_ n\to \mu\) weakly.