The index, the UK's leading measure of share prices, has been rising steadily since the middle December.
It marks the eighth successive record high. The latest boost came from a 1% drop for the pound against the dollar.
A weaker pound boosts the profits of the many multinational companies listed on the FTSE 100 when their foreign earnings are converted into pounds.
The pound's sharp fall was provoked by comments on Brexit made on Sunday by the Prime Minister, Theresa May.
She rejected the idea that the UK could "keep bits of membership" of the EU, raising the prospect of a so-called "hard Brexit".
That sent the pound down against a broad range of currencies.
The pound fell more than a cent, or 1%, against the dollar to $1.2170, and also dropped nearly 2% against the euro to €1.15.
"The pound has dipped to its lowest level since October on Monday after Prime Minister May said on Sunday that we couldn't keep bits of the EU, fuelling fears of hard Brexit," said Kathleen Brooks, research director at City Index.
"On Monday she said that she doesn't accept the terms 'hard' and 'soft' Brexit, and declined that her comments yesterday were a change of stance. This back-pedalling has done nothing to reverse the decline in the pound," she said.
The biggest risers were Glencore, Randgold, BAT, Imperial Brands and Sage group.
Bovis Homes shares were up 1.4% after the firm announced that its chief executive, David Ritchie, was to step down.
Last month, the company warned that building delays in the run-up to Christmas would hit profits.
Among individual shares, William Hill shares fell 1.6% after it warned profits would be at the low end of expectations.
It said a run of "customer-friendly" results at the back end of last year in football and horse racing had cut profits by about £20m.
As a result, operating profits for 2016 were about £260m, compared with previous guidance of £260m-£280m.
Share prices in London rose to a new record high point, with the FTSE 100 index closing 28 points up at 7,238.