My first suggestion would be to not move it, but instead mark it as deleted. This could either be a "status" field which can have different values to indicate what status applies, including, for example, 'D' for deleted. Or you could just have a separate column only for indicating when it is deleted. This would mean you would update your regular listing queries to include a check of that column's value, e.g.:
. . . WHERE status != 'D' . . .
Why I like such an approach is that it makes it easy to generate reports and such since all the historical data is in that one table, plus I just don't like having more tables than I really need.
If you do want/need to go ahead with the idea of moving it to another table, probably the simplest approach is:
INSERT INTO new_table (col1, col2, col3)
SELECT col1, col2, col3 FROM old_table WHERE col1='foo'; -- assuming for this example `col1` is the primary key
Obviously you want to do that before you then delete that record from