||I'm using ArcMap 9.1, so my description is for that version. You can add the dwg file just as you would a shapefile. ArcMap will break up the drawing into the different vector formats - points, lines, polygons, etc. A drawing contains all of these features in one format, whereas shapefiles only allow you to store one type of feature. Then in arcmap, you can right click on the layer that you need, --> data, --> export data. This should let you export to a shapefile. Then you can assign a projection to the shapefile.
Another option if you have access to Autocad Map, or autocad map 3d is to open the drawing there. And use map's export function to export to a shapefile. This option gives you more control over what to export, so you don't have as much clean-up.
Hope this helps,