Alas, this is not as simple a question as it sounds! What sort of accommodation are you planning to use? Could you bear one or two chilly nights when you have to wear all your clothes? How much weight are you planning to carry on your back? ..... use 'search' to discover many opinions on this topic!!
Last year I was on the vdlp in june and some days were bloody cold! In Castro Dozon (about 800 metres) people slept in their sleeping bags and clothes with a blanket over them-those who missed out on the blankets had a sleepless night. I recall at least 2 albergues where they lit the open fires because it was so cold.I used a Snugpak travellite-about 900 gr which was adequate.
I know some of us tolerate the cold better than others. I am one of the others. I was cold many times on the Via last year in May, June and early July. One night it snowed on the mountain in front of the albergue and during the day it was soooo windy, that I wrapped my sarong around my head and neck. Layers is the key. Be prepared for anything.
Peaceable Projects Inc. is a U.S.-based non-profit group that brings the vast resources of the wide world together with the ongoing needs of the people who live, work, and travel on the Camino de Santiago pilgrim trail network in Spain.