jmc

Let the two integers be $x$ and $1998-x$. The product which needs to be maximized is $1998x-x^2$. Completing the square results in $-(x-999{)}^2+999^2$. Since $-(x-999{)}^2\le 0$, the expression is maximized when $x=999$, which results in a value of $999^2=\boxed{998001}$.