Antoine, Benoît, Claude, Didier, Étienne, and Françoise go to the cinéma together to see a movie. The six of them want to sit in a single row of six seats. But Antoine, Benoît, and Claude are mortal enemies and refuse to sit next to either of the other two. How many different arrangements are possible?
1. **Identify the constraints**: Antoine (A), Benoît (B), and Claude (C) cannot sit next to each other. This means that no two of A, B, and C can be adjacent in the seating arrangement.

2. **Determine possible seatings for A, B, and C**: To ensure that A, B, and C are not adjacent, we need to place them in such a way that there is at least one seat between any two of them. The possible sets of seats for A, B, and C are:
   - {1, 3, 5}
   - {1, 3, 6}
   - {1, 4, 6}
   - {2, 4, 6}

3. **Count the arrangements for A, B, and C**: For each of the above sets, A, B, and C can be permuted among the three chosen seats. There are \(3!\) (3 factorial) ways to arrange three people in three seats:
   \[
   3! = 6
   \]

4. **Count the arrangements for D, E, and F**: The remaining three seats will be occupied by Didier (D), Étienne (E), and Françoise (F). These three can also be permuted among the remaining three seats in \(3!\) ways:
   \[
   3! = 6
   \]

5. **Calculate the total number of arrangements**: There are 4 possible sets of seats for A, B, and C. For each set, there are \(3!\) ways to arrange A, B, and C, and \(3!\) ways to arrange D, E, and F. Therefore, the total number of arrangements is:
   \[
   4 \times 3! \times 3! = 4 \times 6 \times 6 = 144
   \]

Conclusion:
\[
\boxed{144}
\]