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The 32-year-old TV presenter had been due to co-present coverage of the 2021 song contest's semi-finals on BBC Four alongside Scott Mills and Chelcee Grimes on Tuesday 18 May and Thursday 20 May.
Clark-Neal will now be replaced by Radio DJ Sara Cox.
The 46-year-old revealed on her her Half Wower show on BBC Radio 2: "I’m really excited, I’ve got a new little job this week. I will be co-hosting the Eurovision semi-finals on BBC Four tomorrow night and Thursday night.
"Rylan is unwell, so obvs we wish him better. So I’ll be stepping into Rylan’s sparkly tux tomorrow and Thursday, 8pm, BBC Four, all the action from the Eurovision semi-finals.
"I’ll also be rekindling the flames of fun with my old Eurovision mucker Scott Mills, and we’ll be joined by Chelcee Grimes and many guests as well."
Anita Rani will stand in for Cox on her Radio 2 show.
Cox previously presenting Eurovision semi-finals coverage with fellow Radio DJ Mills in 2011 and 2012.
Clark-Neal will appear with Mills and Grimes in a pre-recorded sketch parodying hit BBC police drama Line Of Duty.
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The Ready Steady Cook presenter plays a police officer grilling the UK's 2021 Eurovision entry James Newman about his plans for the contest, which takes place in Rotterdam in the Netherlands this year.
The sketch will air as part of the semi-final coverage on Tuesday evening.
Graham Norton will host the finals of the 65th Eurovision Song Contest on Saturday 22 May on BBC One.
The comedian and chat show host has presented the BBC coverage of the contest since 2009.
Eurovision did not take place in 2020 due to the coronavirus pandemic. The 2019 contest saw the UK's Michael Rice placed last, after being awarded just 11 points for his song Bigger Than Us.
Rice, 23, blamed Brexit as the reason the UK was placed last.
He said: “I always knew I was going to come in this position because of Brexit.
“Do you know what? If it was Gary Barlow or Elton John, they still probably would have come last, too.”
The first semi-final of the Eurovision Song Contest 2021 will air on BBC Four at 8pm on Tuesday 18 May.
Watch: Australia will perform at the Eurovision Song Contest via video link