The treatments already described will make the gutters watertight for another year, but if metal guttering is severely rusted you will have to replace the affected section, or even the whole system. If this is necessary, choose a plastic system as it will be cheaper and easier to handle than metal. When buying a replacement section of guttering always saw off a piece of the old system and take it with you to ensure you get the right shape and size. Metal gutters are heavy so get someone to help you fit the new section.
Uncoupling old joints is often easier said than done since the bolts holding the system together are likely to be locked solid by years of rust. Don’t waste too much time trying to force a stubborn bolt but apply a little penetrating oil and try again the following day. If the bolt still will not move, saw stop end outlet through it with a hacksaw. Prise the sections apart and using an old chisel scrape off the sealing material in the joint.
Clean up and treat any rust spots on the adjoining sections. The new section of guttering may have ready drilled holes at each end to take the bolts. If not, support the gutter on a thick piece of wood laid on a flat surface and drill the holes where required. Unless you buy the gutter ready-painted apply a rust-resistant primer inside and out before painting. Use nails to line up the fixing holes in the old and new lengths, spread metal putty into the joint and press the new section into place, wiping off any excess putty with a cloth. Once the new length is firmly seated, insert and tighten the bolts.