I play with a 1950's Titlist, converted for me, by Mike Pancerny of Detroit.
If I remember correctly, he said it was 1.24.
All of my other cues are a millimeter and up to a few millimeters thicker. I have an A.E. Schmidt Titlist conversion that I bought in 1973 that has to be 1.33 or better. It is so huge, it won't fit into my tube case. I carry it in a box case that I bought when I got the cue.