The thief took thousands of pounds after breaking into the home of the woman, who is in her 80s.
Her family told South Wales Police detectives that some of the cash had been intended to pay for her own funeral.
Police said the break-in at Llandraw Court, Maesycoed, Pontypridd, occurred between 3.30pm on August 23 and 11.30am the following day.
Detective Sergeant Nick Jenkins urged members of the public with any information to do the right thing and come forward.
"A significant amount of money has been stolen and somebody somewhere knows something. We believe it is most likely that the person responsible is relatively local to the area," he said.
"On behalf of the woman's family, I would personally like to appeal for anyone who suspects they know who was responsible to do the right thing and come forward.
"That person has a difficult decision to make, but they should imagine this happening to their grandmother, mother or anybody who is dear to them to appreciate how important it is.
"The victim wasn't in the house at the time and the family have decided not to tell their mum what happened so she can focus on getting better.
"But there will come a time when they will have to break the news and it is set to devastate her."
Anyone with information is urged to call South Wales Police via 01656 655555 or Crimestoppers anonymously on 0800 555 111.
Police have confirmed that forensics will be "one line of inquiry" they are examining.
Detectives have already dusted down the property for fingerprints as well as checking for DNA traces before cross-referencing them with a list of convicted criminals already on their database.
They have also carried out door-to-door inquiries, as well as speaking to several local informants.
The crime has shocked even some of South Wales Police's "toughest and most experienced" officers.
It is believed nothing else was taken during the break-in.
In the aftermath of the crime, officers say security at the affected property has been stepped up and beat officers will monitor the home, to deter any would-be thieves as well as to reassure stunned locals.
A force spokesman said: "The woman's family confirmed the money stolen was their elderly mother's entire life savings, which in part was intended to avoid burdening the family with the cost of her funeral one day."