Mills and Clark lost a straight battle for gold in the women's division, heading into the medal race level on points with New Zealand. But the British duo struggled to find the wind and ended up finishing ninth while Jo Aleh and Olivia Powrie took the win and the gold medal.
Lisa Westerhof and Lobke Berkhout took bronze after finishing sixth in the medal race only one place behind France, who needed at least another boat between them and the Dutchwomen to snatch the final medal.
“We just feel a bit gutted at the moment; we didn’t even put a good show on really,” Clark said.
“We had the Kiwis on the start line and we let them go to the right and we were wedged in with a few boats and didn’t follow them out there. Halfway up the beat we knew the gold medal was gone from us.
“When we started sailing together 18 months ago we put everything in to try to qualify for the team. At that point in time we weren’t even thinking about a medal it was just about qualifying. So to come away with a medal together, we’ve had a wicked time.”
Mills and Clark appeared to get the best start but found themselves sharing wind with the rest of the fleet while the slow-starting Kiwis were able to tack right and pick up some clear wind.
The Brits then ended up stranded on the left side of the course as the changeable, light wind favoured the New Zealanders over on the opposite side.
“The wind was tricky,” Mills said.
“We felt like the left had better breeze, which is why we wanted to get the left, but a 20 degree right wind shift came in, the wind died [on the left] and that was game over. We’re pretty gutted to be honest but we’ve had a wicked 18 months together. Sas is amazing and we’ve had a great time.”
Overall leaders and pre-event favourites Mathew Belcher and Malcolm Page, triple world champions, edged a tight contest with their Patience and Bithell who were lagging just behind going into the title decider.
“We tried it, we got the better of them at the start and just managed to sneak ahead but we also got ahead of the rest of the fleet," Bithell said. "They sailed a fantastic race, we tried to attack but they are worthy champions. We’re happy with second, it’s our first Games.”
Patience and Bithell were runners-up to Belcher and Page in the 2011 world championships.
Racing was called off on Thursday because of a lack of wind but the men's event got under way at 12:00 local time, with the women's to follow.
The sailors faced different conditions on Friday with the wind for the first time coming from the southeast but Australia mastered it best to prevail.
“It’s just so good. It’s been three years hard work together, 15 years up there, and here we are standing on the Olympic podium," Patience said.
"We’re young boys, there’s still plenty of time, we will be coming back. What a week, we’ve sailed the regatta of our lives, what a place, we’re in paradise.”
The gold-medal race, which counted for double points, was held on the spectator-friendly, shore-based Nothe course.
The short course, which has gained a reputation for its unpredictable wind shifts, provided the added element of swell coming from the southeast due to the change in wind direction.
The race for bronze saw Argentina's Lucas Calabrese and Juan de la Fuente deny Italy's Gabrio Zandona and Pietro Zucchetti.