* Captured oil increases to 6,000 barrels in a day

* BP moving to improve on that in next few days (Recasts, adds quotes, capacity of containment device)

MIAMI, June 5 (Reuters) - BP Plc BP.L has increased the amount of oil it is collecting through a containment cap fitted over its ruptured Gulf of Mexico oil well, and is working to improve on that rate, the U.S. national oil spill response commander said on Saturday.

U.S. Coast Guard Admiral Thad Allen told reporters the containment effort was working “with caveats”, although it was still only collecting part of the leaking oil at this stage.

“In the first full 24-hour cycle yesterday ... they were able to bring up ... 6,000 barrels of oil from the well,” Allen said, speaking in Theodore, Alabama.

Using the government’s highest estimate of 19,000 barrels (800,000 gallons/3 million litres) per day spewing from the blown-out Macondo well, the 6,000 barrels captured on Friday would represent nearly a third.

While he avoided giving a percentage of how much of the escaping oil was being captured, since BP, the government and outside experts have provided different estimates of the size of the leak, Allen stressed that the amount collected over 24 hours was an improvement over the 1,000 barrels per day (bpd) reported being collected previously by the containment cap.

“They started out at a lower rate and are going to a higher rate,” he said.

He said the full capacity of this containment device was 15,000 bpd, which he called the “upper limit” of what can be done with this attempt to control the leak.

“On June 4, a total of 6,077 barrels of oil was collected. Improvement in oil collection is expected over the next several days,” BP said in a Twitter feed.

Allen said the company hoped to raise the amount being collected in the next few days by closing vents in the containment cap as engineers managed to stabilize pressure in the gushing well.

BP plans further improvements in the containment operation this month, although hopes for a final solution to plug the well are fixed on two relief wells being drilled that will be completed in August. (Reporting by Pascal Fletcher and Chris Baltimore; editing by Philip Barbara)