That can be fixed, but personally I wouldn't leave it to just any welder. It's best to get in touch with a frame specialist.
And yes, the paint will need to be (partially) stripped, so you're looking at a complete strip and repaint as well if you decide to fix it.
Getting a new frame will probably be the cheaper option, but I personally feel that it would be a waste to scrap a frame for such a minor crack.
It's a classic after all and as you mention it's a trusty steed, so you probably have some fond memories of it. IMO that justifies the extra cost of getting it fixed up.
Oh, and I know it's dangerous to describe any crack as "minor". You are not safe on that bike anymore until it's fixed, so it really needs to be looked at before you get on it again.
I'm calling it minor because it's not opening up or going all the way around. At this point it's easy to fix.