jmc

There are 6 choices of seats for Fred to sit in. Once Fred is seated, then Gwen must sit opposite him. This leaves 4 people to place in the four remaining seats, which can be done in $4!$ ways. However, we must divide by 6 to account for the 6 rotations of the table. So the number of arrangements is $\frac{6\times 1\times 4!}{6}=4!=\boxed{24}$. Alternatively, we could start by fixing the table around Fred, thus removing the rotation. There is 1 option for Gwen's seat, since she must sit across from him. This leaves 4 people to place in four unique seats, so the number of arrangements is $4!=\boxed{24}$.