The 22-year-old won his first medal in the 10m platform event, finishing with a score of 543.80.
China's Olympic champion Aisen Chen took gold with 580.60 while silver went to compatriot Hao Yang (574.80).
Daley then teamed up with 20-year-old Grace Reid to finish third in the mixed 3m synchro behind pairs from China and Canada.
Their best dive came in the third round when they scored 69.30 points for their inward two-and-a-half somersaults piked.

The satellites were launched on a Soyuz rocket from French Guiana.
Their successful placement in orbit brings the number of platforms in the constellation to 10 - a third of the way to a full network of 30.
Lift-off occurred at 02:08 GMT, with confirmation of the satellites' separation from the rocket's upper-stage coming almost four hours later.
Galileo is a project of the European Commission, the EU's executive branch.
It is designed to complement the American Global Positioning System (GPS), while at the same time introducing extra precision for users.
Although dedicated services based on the European network will not be made available until at least 14 operational satellites are in orbit, the signals from every new addition in the sky can be exploited by receiving devices with compatible chipsets.
"There are a number of chipsets that have been developed and are in the market deployed in smartphones and navigation equipment for cars, for instance," explained Javier Benedicto, the Galileo programme manager at the European Space Agency (Esa is the EC's procurement agent).
"Those chipsets are already able to combine the Galileo signals with the GPS signals. That converts into an improvement of the availability of the navigation service which is experienced by GPS users today."
The development path to a "European GPS" has been a tortuous one.  The project is years late, and the completion cost - expected to be some â‚¬7bn by 2020 - is substantially higher than that originally foreseen by EU member states.
But Didier Faivre, the director of navigation at Esa, said the delivery cadence agreed in 2013 was being maintained.
"The famous '30 satellites by 2020' motto is our goal, and we're confident we can do it with additional procurement initiated this year," he told BBC News.
The satellites for Galileo are being made by a German-UK consortium. OHB System of Bremen is the industrial prime contractor and assembles every spacecraft bus, or chassis.  The navigation payloads, including the atomic clocks that are at the heart of any sat-nav concept, are prepared in Guildford by Surrey Satellite Technology Limited.
The consortium is now churning out two satellites every three months.
A pair is currently waiting to be shipped to French Guiana's Kourou spaceport from Esa's test centre in the Netherlands for another Soyuz launch in December. A further duo (these would be numbers 13 and 14) will soon enter final testing before also being shipped to Kourou.
Esa plans just one launch in 2016, sending up a quartet of satellites on the much bigger Ariane 5 rocket. The year 2017 would likely see two launches - one pair of spacecraft lofted by Soyuz, and another quartet on an Ariane.
To date, the EC has only ordered 26 satellites, so it will need soon to order more if it wants to attain the magic number of 30. However, the next procurement will almost certainly call for more than four platforms because of problems with three spacecraft already in orbit.
It is not clear yet whether this trio will be able to take their place in the final constellation.  One had a power failure and can no longer broadcast on all its frequencies, and the other two were put in an incorrect orbit by their Soyuz rocket.
Engineers plan to make changes to Galileo's ground systems to take account of these inadequacies, but it is by no means certain that these modifications will allow the degraded satellites to assume a fully functional role in the network.
Jonathan.Amos-INTERNET@bbc.co.uk and follow me on Twitter: @BBCAmos
Britain's Tom Daley won two bronzes on the final day of the Diving World Cup event in Kazan, Russia.
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Europe has put up another two spacecraft in its Galileo sat-nav system.