Yulia Solodyankina, a 22-year-old physics undergraduate at Edinburgh University, was last seen in the capital before boarding a bus for Glasgow on June 7 last year. CCTV footage captured her at Buchanan bus station at 4.55pm and she was not seen again.
The remains were discovered off a track near the A83 near the village of Arrochar in Argyll on January 9. They were badly decomposed and her identity was established following forensic examination and DNA testing.
The police statement said there were no suspicious circumstances surrounding her death and her family had been informed.
A report has been submitted to the procurator fiscal.
Friends who set up the Find Yulia campaign on social networking websites last night paid tribute to the young woman.
They said: "No words can express the sadness we feel regarding the death of our friend Yulia. We feel blessed to have spent the time that we did with her. Yulia was full of energy, a beautiful dancer, caring friend and a kind and gentle soul.
"Our thoughts go out to her family and loved ones. We would like to thank the thousands of people involved in the Find Yulia campaign.
"We feel overwhelmed by the number of people who offered their support and sent kind words. It has been heartening to realise how many people really care. We would encourage anyone struggling or feeling a sense of desperation to reach out for help and support. We miss Yulia so much and will never forget her."
More tributes poured in to the site, with many people passing on their condolences to her friends and family.
Ms Solodyankina attended a concert at The Wee Red Bar in Edinburgh the night before she disappeared, but left saying she felt unwell. The next day she left her flat in Edinburgh city centre, leaving behind her wallet and mobile phone, before heading to Glasgow. Her boyfriend, Antoine Dao, said she had been "very worried" about the results of her final-year examinations.
Ms Solodyina's parents, Ekatarina and Dmitry, made impassioned pleas for their daughter to return home. A Facebook post at Christmas said: "We are still looking for Yulia. It has been over six months since she went missing and her friends and family have not forgotten her. We would all love to know that she is safe and well."