It sounds like you are trying to project from NAD83 UTM Zone 12 to WGS84 UTM Zone 13N (presumably you are working on/near a zone boundary).
At least in 10.1, I found WGS 1984 UTM Zone 13N under Projected Coordinate Systems>UTM>WGS84>Northern Hemisphere and I would expect it to be found in the same location at 9.3.
So I think you should be able to do the projection you want in a single step using the Project tool.
Whenever you have to transform or convert data between two geographic coordinate reference systems (GeoCRS, also called datums), you may find that there are zero, one, or multiple possible transformations available.
If there are multiple transformations, they usually differ by area of usage and/or accuracy. Some countries like Belgium have published several transformations that cover the same area but with improved accuracy as more data has become available.
For other Geographic CRS, there may be multiple ones for different areas, plus some with different accuracies. Some places to find out what's available include spatialreference.org and epsg.io. Please be aware that a good portion of the information at these two websites originates from the EPSG Geodetic Parameter Dataset, hosted by IOGP (International Association of Oil and Gas Producers).
In this case, the area of interest is in Barcelona - Catalonia. Checking for that area and transformations between ED50 and WGS84, finds ED50 to WGS 1984 (28), EPSG::1633 (in Esri terms, ED_1950_To_WGS_1984_28) which was originally published by CNIG and has a listed accuracy of 1.5 m.
Disclosure: I'm on the Geodesy subcommittee that maintains the dataset, and I work at Esri.
Best Answer
If you are planning to perform spatial analysis using your data, I suggest you to reproject all your data into one projection, either using ArcGIS or FME. Sometimes the reprojections does not work perfectly on the first try, you may need to try a couple times to get it right.