The Queen has been informed of the news.
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Estimate, a five-year old mare, won the Gold Cup at last year's Royal Ascot - one of the most respected racing meets in the world. This year, the Queen's horse came second.
While its Gold Cup victory is not under threat, the £80,625 ($137,458) runner-up prize money from this year's event may have to be forfeited as a result of the positive test.
The horse's trainer Stoute is unlikely to face any penalty if the investigation finds that the morphine was ingested inadvertently.
Morphine is a naturally occurring substance found in flowers like poppies. It can contaminate growing crops, and as such, end up in animal feed.