The patient died at the Shrewsbury and Telford Hospital NHS Trust (SaTH) on Thursday and their death was announced by the NHS on Saturday.
It is not known whether they were being treated at Princess Royal Hospital in Telford or Royal Shrewsbury Hospital.
It is the first time a Covid-19 patient has died in hospital in Shropshire since July 12 and takes the death toll in the county's hospitals to 190.
No new deaths were announced in Shropshire on Sunday.
A total of 170 patients have died at SaTH, 15 have died at Shropshire Community Health NHS Trust, which runs community hospitals in Ludlow, Bridgnorth, Bishop's Castle and Whitchurch, and five have died at the orthopaedic hospital in Gobowen.
A further 156 patients have died in care homes in the county after contracting coronavirus, with the most recent care home death on September 16.
Of these 106 were in homes in the Shropshire Council borough and 50 were in Telford and Wrekin.
The NHS confirmed 42 new deaths in hospitals in England on Saturday and 28 on Sunday, taking the total hospital death toll to 30,166.
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The UK-wide death toll released by the Government, which includes deaths in and out of hospital, increased by 49 to 42,317 on Saturday.
Meanwhile 68 new positive tests were reported in Shropshire and Telford and Wrekin on Saturday, although the Government said the number of cases being reported would increase sharply due to a technical issue with reporting.
A delay in publishing the number of Covid-19 cases in England means the total reported over coming days will include extra cases that should have been reported between September 24 and October 1.
It comes after public health officials across the county urged local residents to play their part in preventing the spread of coronavirus as the number of new cases continues to rise.
The latest NHS data shows the infection rate in the Shropshire Council was 33.4 cases per 100,000 people in the week to September 30 when 107 new cases were confirmed.
In Telford and Wrekin the rate was at 32.1 per 100,000 people, after 57 new positive tests.
The highest infection rate in the West Midlands is in Birmingham, at 93.1 cases per 100,000 people, while Manchester and Liverpool have the highest infection rates in England at 290.2 and 278.9 respectively.