That is what I suspected.
Here is what you, or an electrically knowledgeable person, needs to do from the info you gave:
First - do not do wiring alone. Have someone else there to be your safety man and maybe hold a flashlight.
1) Shut OFF the two breakers that have the black and white wire connected. If not comfortable working in a fuse box, shut off the main breaker and work with a good flashlight. Once the wires have been moved in the fuse box you can turn on the main breaker ... but leave the two breakers you are working on OFF.
2) Remove the old 220 socket in the garage and replace it with a NEMA 6-20 or L6-20. See check list below for the proper wire connections.
3) Remove the white wire from the breaker it is attached to. That breaker is now a spare 110V breaker for a new circuit someday. NOTE- the breakers should be separate and not tired together.
4) Connect the white wire to the neutral bus where all the other wires are. You may have to use a wire nut and splice on some extra white wire if it is not long enough.
5) Double check the wires as listed below and turn on the breaker. Check the receptacle with a meter to assure a proper circuit.
6) Depending on the plug on your welder, you may need to change the plug to match the receptacle, or make an adapter cord.
Checklist to verify it is now wired for 115V-20A:
____ 20-amp single pole breaker in breaker box
____ 20-amp receptacle for 115V (NEMA 6-20 or equivalent) on wall
____ Fuse Box - Black wire on breaker
____ Fuse Box - white wire on neutral bus
____ Fuse box - green wire or bare copper wire on ground bus (in some fuse boxes this is the same as the neutral buss)
____ Receptacle - black wire on the brass color screw
____ Receptacle - white wire on the silver color screw
____ Receptacle - Green/bare wire on ground lug - NOTE -If this wire is not there you have a problem and will need to run a new wire.