Hi Cloé,
My wife and I walked the Francés in mid-April to late-May 2010, and the weather was variable, to say the least. Walking over the Pyrenees we had glorious sunny weather, mid-teens (celsius) and perfect for walking. A few days later, we had dense fog and light rain and mud climbing Alto de Perdón. The weather while we walked the meseta was generally good, but from about Astorga on, it got chilly (due to the altitude, I think) and going over O Cebreiro, and for the following 2 or 3 days we had snow, sleet and rain (and some of the albergues were unheated). Not nice! At one point along there, we pulled out our second pairs of socks for mitts.
I'm not forecasting your weather, obviously, but within the limits of keeping your pack light, IMO you should be at least mentally prepared for a wide range of conditions. Remember, however, that you can always find things, like fleeces or better/different rain gear, in the bigger cities (like Pamplona, Burgos, León, Astorga) and some of the smaller ones, if you need to. Most of all, enjoy your camino, whatever it may bring.
Buen camino!