Constable Rhianydd Hitchcock was found by colleagues at around 7.40pm on Saturday following an incident at Alloa Police Station in Clackmannanshire.
A post-mortem examination will be carried out, however police said there do not appear to be any suspicious circumstances surrounding her death.
Pc Hitchcock, 29, had a one-year-old daughter.
In a statement released through police, her family said: "We would like to thank everyone for their kind words following the sudden passing of our beloved daughter and sister Rhianydd, proud to have been the first Type One diabetic accepted into what was then Central Scotland Police.
"She was accomplished, with so much left to do. Our big lassie will be so missed by ourselves, her special girl Maia and all her many friends.
"Following this senseless tragedy we ask to be allowed to deal with our grief privately at this time."
Davie Flynn, Forth Valley Divisional Commander, said: "Rhianydd was a loyal and committed officer, well respected and popular with colleagues. She was doing the job she loved and had recently gained the leadership diploma, allowing her to be considered for promotion.
"We are very saddened by her death, it is a tragic loss and she will be sorely missed by everyone who knew her.
"Our sympathies are very much with her family at this distressing time. We will do all we can to support them as they try to come to terms with her passing."