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Default Count cells based on date range in another column

Could someone please suggest how to use a worksheet function to count
the number of non-empty cells in one column when the value in another
(date) column is in a particular date range? The first column happens
to be dates as well, but the formula would ideally work regardless of
the counted column's data type.

e.g.

Given data such as:
Column 1 Column 2
--------- --------
23-Oct-03 2-Jan-04
12-Dec-03
1-Jan-05 14-Feb-05
31-Oct-02 12-Dec-03
1-Jan-05 14-Feb-05
31-Oct-02 12-Dec-03

How do I count the non-empty cells in Column 1 when Column 2 contains a
date in the year 2003?

Thanks,

Terry