Please try it by giving a static weight for all products like 0.1 weight for all products by using script.
1) If you want to do it by database then here is the code:
select * from eav_attribute where attribute_code='weight'; it will give you id of that attribute
select * from catalog_product_entity_decimal where attribute_id=82 // find that attribute value
update catalog_product_entity_decimal set value='' where attribute_id=82 and value='0.3'
2) If you want to do it when product save:
Create this event
catalog_product_save_after and load product and update its weight.
This event run everytime after product save.
3) Create a script to do this.
This is the temporary solution. Hope it works!!