"You only believe it is done when you have actually done it. This is the biggest financial reconstruction in the world and there are as many moving parts as you could imagine," he says, refusing to be drawn into predicting that the deal will definitely go through.
"It's all right for you to believe that, but not for me," he adds with a wry smile.
If Mr Rowland does pull it off at the key votes in the summer, as seems highly likely, he will find his reputation transformed. The man who drew the short straw and found himself fighting a desperate three-year rearguard action on behalf of what seemed a disreputable bunch of loss-making insurers will become the hero who kept a pounds 27bn business on its feet against all odds.
Those odds are certainly moving in Lloyd's favour. A Mori opinion poll recently showed a large majority of members are likely to accept the offer of compensation for the disasters of the past.
Straw polls after each meeting during Lloyd's international road shows have given much the same result, says Mr Rowland.
The story of how Lloyd's stopped digging itself ever deeper into a hole of losses pounds 8bn deep will take some unravelling over years to come. One important factor seems to have been Mr Rowland's ability to think instinctively and quickly on his feet, both in public and at private meetings.
There have been moments at mass meetings of thousands of Lloyd's members - the names - in the Royal Albert Hall and elsewhere when the rescue has hung in the balance.
"Until you have experience of handling some of these very big meetings with extremely angry people in them you don't know if you can do it. Nobody can teach you," says Mr Rowland, whose long years as a broker and previous job as chairman of Sedgwick, one of the biggest brokers, had not prepared him for what he had to face.
Observers of his performances praise his adroit reactions and steadiness under fire, a quality noticeable from the opening of his first meeting of members at the Albert Hall.
"I said good afternoon, ladies and gentlemen, and literally at that moment from the gallery a voice rang out - `liar' - and I thought, I haven't said anything yet. But you just have to keep going, to plough on."
Though the Lloyd's saga has been wrapped up in obscure technicalities that only the most dedicated can possibly follow, Mr Rowland identifies the simplest and most obvious of turning points in its fortunes that made the rescue possible: "The most important thing is that this place started making money again. If that hadn't happened we would be nowhere.
"We demonstrated that we had a business that could make returns, and we held our clients amazingly during this period." Lloyd's is expected to make pounds 2.5bn profit for 1993-95.
The combination of more money in the market and the resounding success of Lloyd's names in court actions persuaded even the most grudging member firms that something far bigger than the derisory pounds 900m first offer in 1993 was required to clear the debris of the past. That first offer was rejected, though by a surprisingly narrow margin.
The second significant turning point Mr Rowland identifies was the change in the rules that ended a long tradition, by allowing limited liability investors into the market. There is now pounds 3bn of this corporate capital in a market total of pounds 10bn.
Mr Rowland says: "We showed to the world that a lot of people wanted to invest in this business. This is an infinitely more powerful weapon than sweet words from me."
The limited liability proposal came from a task force report put together by a team under Mr Rowland, while he was chairman of Sedgwick. The task force revealed astonishing ignorance among insurance practitioners of the way their market actually worked.
Even Mr Rowland admits that, despite names' well-known liability down to the last cuff-link, many people including himself did not realise until Lloyd's was threatened with collapse that unlimited liability also meant that while you could resign from the market, you could never quit. Lloyd's liabilities follow you until the end of your days, Mr Rowland says. This is because resigning members reinsure outstanding liabilities with syndicates that remain in the market. If Lloyd's, as reinsurer, were to fail, any claims would revert to the original names.
"The strange thing is how people cloud their minds to some of the reality. I don't think I realised. I thought that if I resigned from Lloyd's, that was it."
Mr Rowland pays fulsome tribute to the team effort involved in rescuing the market. But that team appeared to be breaking down earlier this year when Peter Middleton resigned to go to Salomon Brothers, an unwelcome surprise to Mr Rowland. Mr Middleton was the blunt-spoken chief executive who had run almost a double act with the urbane Mr Rowland, and who had shown a remarkable ability to get close to the angry names.
In a polite rebuke, Mr Rowland says: "I just happened to think that you should not bail out of something as important as this. I told him that. It was absolutely my mistake to assume other peoples' motivations are the same as your own. Peter saw this as a job and was offered what he considered a better job - fine! Other colleagues have a certain passion and commitment to what is happening that goes beyond that." Luckily, Lloyd's had recruited a potential successor in Ron Sandler, and no lasting damage was done.
At this point the differences of motivation between a professional manager such as Mr Middleton and a passionately committed insurance man who has spent his entire life in the business become stark.
Mr Rowland says: "My motivation is quite different [from Peter's]. I have been 37 years an insurance broker - not in the market, but of the market. Of course, I have lots of strong feelings about it. It would be absolutely disgraceful in the truest sense of the word to have allowed Lloyd's to destroy itself or to be destroyed."
Even to the last, suspicion and bitterness run deep among members. For example, Mr Rowland rejects allegations that Lloyd's was playing games with the figures this year, suddenly finding an unexpected pounds 1.2bn to top up the offer. The market's contribution was raised from pounds 2.8bn to pounds 3.1bn and the cost to names of financing Equitas, the reinsurance vehicle that will ring-fence past liabilities, plummeted by pounds 900m to pounds 1bn in the space of two months.
Perhaps the firmest sign that Lloyd's really believes it will get the offer through is that Mr Rowland is already planning to spend his next and final year as chairman selling the market's services world-wide.
As for the executive team, he has has been telling them: "If you think when we are through this we are going to have a holiday, I think we will have 24 hours off and then get down to the real job."