Pasting problem with hidden/filtered cells

I am having a problem that deals with pasting on a filtered column.

I have this big spreadsheet and I have to add changes to it that I get from another spreadsheet. What I have done is put auto filters on the column headings. Then I filtered to only show those that have start dates. What I would like to do is paste in the column of changes into a column on my original spreadsheet. When I try to do this, the changes ended up being pasted in both the visible and hidden cells, which messes everything up.

Can anyone help me figure out how to paste only in the visible cells?

I am using Excel 2003.

THANK YOU!