The surprise appointment of Mr Garraway, 60, comes after recommendations earlier this year by Sir Jeremy Morse, the leading banker, that Lloyd's needed to reform its system of government. Sir Jeremy wanted two bodies created - one specifically for developing the business of the market and the other for regulating its affairs.
Mr Garraway is to assume the two-days-a-week role after Andrew Large, the head of the Securities and Investments Board who sits on the Lloyd's ruling council, indicated that he was unable to take on the task due to pressure of work.
In his post, Mr Garraway will sit on Lloyd's ruling council and also have the courtesy title of deputy chairman. In the hierarchy at Lloyd's he will be more senior than the recently appointed Peter Middleton, who has become chief executive officer of the new board responsible for developing Lloyd's business.
Mr Middleton, who headed the travel agent Thomas Cook, is earning pounds 250,000 a year at Lloyd's. Mr Garraway, who was earning more than pounds 500,000 at BAT, is in discussions with Lloyd's about his fees.
Mr Garraway will be one of three deputy chairmen, the other two traditionally drawn from the market's professionals. Mr Middleton said yesterday that the role of the deputy chairmen in the new structure had yet to be worked out.
Mr Garraway has no experience of the regulation of insurance but has been actively involved in BAT's expansion into the field.
He helped the group to acquire the insurance group Eagle Star, the US Farmers Group and Hambro's (now Allied Dunbar). But he has had to deal with extensive leadership and management problems at Eagle Star and Allied Dunbar, which led to many senior departures.
As part of further attempts to help underwriting members meet the record pounds 2bn worth of losses that have flooded into the market, Lloyd's said it only intended to seek recovery of some of the money over a three-year term. But this move would not affect Lloyd's retaining a charge over a member's principal residence and other rights of recovery.
Mary Archer, wife of the novelist Lord Archer and a council member, who has been looking after hardship arrangements, said she had received appeals for help from 1,582 members.
A special support fund of pounds 50m is being set up, using contributions from brokers and underwriters working at Lloyd's.