Sounds a bit like a homework assignment ;-)
A couple of tips:
1. Are you sure both files are each in sorted order (if you can assume "sorted order", it definitely simplifies your "merge" algorithm)
2. I think the main loop should probably look more like this:
while (!File1.eof()) && !File2.eof())
<= I'M NOT SURE YOU NEED TO PUT YOUR READS IN SEPARATE, NESTED LOOPS
3. You might be able to get away with reading a whole line at a time (instead of parsing each line into separate variables), then just doing a simple string compare.
4. The basic algorithm (assuming item "1" above) is to read a line from one file, then keep reading lines (and writing) lines from the second file until the next line is "greater" than the line from the first file. Etc...
'Hope that helps .. at least a little bit ..