I have a different cross stitch die but the tips I have should work the same.
I would use 110 lb cover (297-300 gsm. Make sure it says this on the packaging/listing, otherwise it'll be 110 lb. index weight which is 200 gsm. For reference 80 lb cardstock is about 216 gsm) cardstock. You can go lighter but I want a heavy weight so the paper is less likely to rip if I'm using more threads and can withstand a bit of tugging.
Although with thicker paper you might end up getting that rectangle plate the holes are on pressed into the paper as well. In that case I'd cut 2 papers separately and glue them together. The top layer can be thinner like 65 lb which should be thin enough that it wouldn't get the plate impression and the base layer can be heavier.
If you can't find heavier paper, just glue together sheets of whatever cardstock you have. 65lb is 176 gsm but if you glue 2 sheets it's a little more than 110 lb cardstock and the glue stiffens as it dries giving it more rigidity. I've used 6 sheets of 65 lb cardstock and a glue stick to make really sturdy notebook covers since I didn't have heavyweight chipboard on hand.
I use a glue stick as it has less moisture than liquid glue so there's less chance of warping but it dries hard and is not tacky which tape would still be. It also gives me a bit of time to make sure the paper is lined up evenly. Then I sandwich it between 2 sheets of lightweight chipboard (cut up cereal boxes) and roll it through my die cutting machine to make sure it's pressed tight together. If you have new unmarked plates, that would work but since I don't I used the cereal boxes to protect the paper and double as a shim.
A benefit to the gluing 2+ sheet method is that there are a lot of different color cardstocks and papers readily available these days including foil and shimmer. You can have that top layer be foiled cardstock, designer scrapbook paper, or even a project life card and the bottom be plain cardstock you don't mind covering up. Despite the die having a bigger hole count, it's a good alternative to perforated paper for small projects since that can be very expensive.
By: Bright Eyes
(07-Jun-21)
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