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UPDATE 6-Oil rebounds but still threatened by growing U.S. supply

* U.S. drilling activity continues to rise

* OPEC's supply cuts undermined by higher U.S. output

* Iran's output surpasses 3.8 mln bpd

* Brent prices heading for near 20 percent fall in H1 (Adds quotes, updates prices; changes dateline from LONDON)

NEW YORK, June 26 (Reuters) - Oil prices rebounded on Monday after last week's seven-month lows, but were hemmed in by a relentless rise in U.S. supply and a surge in demand for short sale contracts that signal investors see potential for a price fall.

Brent crude futures were up 25 cent, or 0.6 percent, at $45.79 a barrel by 12:03 p.m. (1603 GMT), still set for a near 20 percent drop in the first half of the year.

U.S. crude futures were up 35 cents, or 0.8 percent, at $43.36 a barrel.

Investors in U.S. crude futures and options, however, increased their bets against any future further rise in prices, as the number of U.S. oil rigs in operation hit its highest in over three years.

U.S. production could jump to 10, maybe 10.5 million barrels a day by the end of the year, and when you add Libya, Nigeria and North Sea production that will negate the Saudi-led cuts," said Gene McGillian, manager of market research at Tradition Energy in Stamford, Connecticut, referring to U.S. output which has steadily grown to around 9.35 million bpd.

The rise in supplies threatens efforts by the Organization of the Petroleum Exporting Countries and its partners to reduce global oil inventories with production cuts.

OPEC states and 11 other exporters agreed in May to extend cuts of 1.8 million barrels per day (bpd) until March, in the hope that it would force global supply and demand to align.

However U.S. shale oil output is up around 10 percent since last year, while Nigeria and Libya, who are exempt from the OPEC cuts, have also hiked output.

Iran, which was given a small increase so it could recover market share lost while under Western sanctions, said its production has surpassed 3.8 million bpd and is expected output to reach 4 million bpd by March.

U.S. drillers have added rigs for 23 straight weeks, energy service company Baker Hughes said on Friday.

"The perception is that were going to have slower exploration activity, but the amount of drilling that weve been doing is going to guarantee production growth for at least another four or five months," said James Williams, president of energy consultant WTRG Economics in London, Arkansas, "So youve still got the downward pressure there."

Analysts at Bank of America-Merrill Lynch said demand had not grown quickly enough to mop up any excess output.

"Looking into the second half of 2017, we now doubt that demand growth will accelerate sufficiently," they wrote. (Additional reporting by Amanda Cooper in London, Jane Chung in Seoul; Editing by Marguerita Choy and Edmund Blair)