We consider deformations of the singular `global cosmic string' compactifications, known to naturally generate exponentially large scales. The deformations are obtained by allowing a constant-curvature metric on the brane and correspond to a choice of integration constant. We show that there exists a unique value of the integration constant that gives rise to a non-singular solution. The metric on the brane is dS<sub>4</sub> with an exponentially small value of the expansion parameter. We derive an upper bound on the brane cosmological constant. We find and investigate more general singular solutions - `dilatonic global string' compactifications - and show that they can have non-singular deformations. We give an embedding of these solutions in type IIB supergravity. There is only one class of supersymmetry-preserving singular dilatonic solutions. We show that they do not have non-singular deformations of the type considered here.