Anyone convert from oil to natural gas heat?

With heating oil close to $4 gal I need to do something. I live in a brick home built in the 50's. Just brick with lath and plaster. No insulation in the walls and no place to put it.

I do have natural gas by the house. I know this may be hard to answer but will natural gas save me money? Interested in anyone who may have converted from oil to gas and what Kind of savings (If any) they saw.