We want to move our current gas got water service about 3 metres away on the same exterior wall. Is this hard to do? What is it likely to cost?

It is not "hard" to do for a professional. I highly recommend that you hire a licensed plumber since gas work is required. As far as cost goes - it varies widely and it is set by your location. I suggest you call 3 plumbers from your local phone book and get 3 free estimates. That should give you a general idea how much to expect to pay. Back to you. MIlo

Not only must you replumb the water lines and the gas line, you must also provide for exhaust gas venting.