Twiney, my niece recently had this problem with her 1930's terraced house. Her kitchen stop cock didn't stop. Although it shut completely there was still some flow. No problem I thought, the outside Stop valve can be shut off and a new kitchen Stop cock fitted. The outside SV hole was empty! United Utilities called in the correct situation - which they did: dug up, new valve with metal cover, and re-tarmaced. It's their water SV after all. Took about three weeks though.
If you don't manage to get them [water board] to resolve your problem, get a new stop cock fitted, above the old one if nec, whilst the pipe is frozen.
Someone recently told me to never open stop cocks fully, but leave with about one turn remaining - allows the threads to free up more easily several years later. Seems sensible to me.
Would I make the new connection through the frozen section b4 it or after it? Will the lead pipe expand if so I might not be able to get the copper to lead compression joint fit would the lead B quite weak over 100 years old!