Yes you can make a dwarf weeping willow tree. But you will need to keep it in a pot with good drainage. Full size trees can be made dwarf by keeping them in a pot, and during the deciduous period you may need to trim the roots. The deciduous time is when the leaves are falling off in the fall.
It is important to make sure the tree is getting a lot of moisture, so they grow best out side even in the winter. I had a branch that was in a pot last year. It survived well over the winter. This year it was what I advertised to bring people to the yard sale. I make willow trees from branches as a hobby.
However if you transplant the tree into the ground "Beware" as you will get a full size tree.
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