When defining projection from ArcCatalog (shapefile properties) I get an error message "Failed to Alter the Spatial Reference". If I try to do this from the "Define Projection" geoprocessing tool (ArcToolbox) I get this error insted: "Cannot create a projection file for filepath Failed to execute (DefineProjection).

I solved this problem by accesing the shapefile through a mapped drive (a.k.a. "Z:\") within ArcCatalog, rather than its \\server\drive$ pathname.