In this paper: Human papillomavirus is a necessary cause of invasive cervical cancer worldwide the authors analysed samples which had previously tested negative for HPV with more advanced methods and found them positive, raising the rate (of HPV in cervical cancers) from 93% to 99.7%.
They state in the abstract that:
> Combining the data from this and the previous study and excluding inadequate specimens, the worldwide HPV prevalence in cervical carcinomas is 99·7 per cent. **The presence of HPV in virtually all cervical cancers** implies the highest worldwide attributable fraction so far reported for a specific cause of any major human cancer.
This means that nearly all cervical cancer cells contain HPV DNA. I don't know if any studies have been done to infect animals with an HPV equivalent to see if they develop cervical cancer. But a correlation of 100% is pretty compelling evidence by itself.