Circumferential cracks in hollow circular cylinders are often used as idealizations of flaws in weldments caused by lack of penetration. The most common cases occur for butt welds in pipes, nozzles, and cylindrical pressure vessels. The objective of the present work is to develop a methodology to compute accurately values of stress intensity factor for the entire range of radius ratio, R_i/R_o (inner to outer), from 0 to 0.9999 and crack-depth-to-thickness ratio, a/t, from 0 to 1.0 for general loading using the weight-function approach. The p-version of the finite-element method was used to obtain stress intensity factors and crack face displacements for the reference loading of uniform tension. The reference solution was obtained for selected values of the geometrical parameters, R_i/R_o and a/t, covering their whole range. Both internal and external cracks were treated. Piecewise cubic Hermite interpolation techniques were then used to compute the quantities corresponding to intermediate values of the geometrical parameters. The derivatives of crack face displacements needed in the weight function method were obtained numerically from a three-term Williams series fit to the displacements obtained from the finite-element analysis. The results obtained were compared to existing solutions for uniform tension loading and excellent agreement was found. Results for uniform tension loading obtained from the weight function method were compared with those from finite-element analysis and error bounds were established.