An ab initio method is described for solving protein structures for which atomic resolution (better than 1.2 A) data are available. The problem is divided into two stages. Firstly, a substructure composed of a small percentage ( approximately 5%) of the scattering matter of the unit cell is positioned. This is used to generate a starting set of phases that are slightly better than random. Secondly, the full structure is developed from this phase set. The substructure can be a constellation of atoms that scatter anomalously, such as metal or S atoms. Alternatively, a structural fragment such as an idealized alpha-helix or a motif from some distantly related protein can be orientated and sometimes positioned by an extensive molecular-replacement search, checking the correlation coefficient between observed and calculated structure factors for the highest normalized structure-factor amplitudes |E|. The top solutions are further ranked on the correlation coefficient for all E values. The phases generated from such fragments are improved using Patterson superposition maps and Sayre-equation refinement carried out with fast Fourier transforms. Phase refinement is completed using a novel density-modification process referred to as dynamic density modification (DDM). The method is illustrated by the solution of a number of known proteins. It has proved fast and very effective, able in these tests to solve proteins of up to 5000 atoms. The resulting electron-density maps show the major part of the structures at atomic resolution and can readily be interpreted by automated procedures.