The Bees will play the tie at Ashton Gate on the 23rd or 24th of August with a 7:45pm kick-off.
Paul Fairclough told Barnetfc.com:
"We will go there strong", promised the Barnet boss.
He added: "The most important thing is that at this time last year we were speculating on a year in the Conference and a draw in the FA Trophy.
"This is another great adventure for me and my young players ... I am grateful to be playing any one of the 71 other clubs in the Football League's Cup competition".
Views from the Messageboard:
"By my reckoning the worst draw we could have got!"
"Look on the bright side, a trip to a decent ground - one that many of our fans won't have been to before. I was there in the FA Cup 10 plus years back, but I haven't been back since. We'll just have to win to get our dream tie. It is going to be tough thou...."
"Not unwinnable"
"there'll be a mediocre crowd and even if we were to win it wouldn't be much of an upset. I'd have much prefered them at home."
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The full draw
Preston v Barnsley
Chesterfield v Huddersfield
Blackpool v Hull
Mansfield v Stoke
Rotherham v Port Vale
Nottingham Forest v Macclesfield
Sheffield United v Boston
Derby v Grimsby
Leeds v Oldham
Wrexham v Doncaster
Scunthorpe v Tranmere
Stockport v Sheffield Wednesday
Bury v Leicester
Rochdale v Bradford
Hartlepool v Darlington
Wolves v Chester
Lincoln v Crewe
Burnley v Carlisle
Reading v Swansea
Gillingham v Oxford
MK Dons v Norwich
Plymouth v Peterborough
Shewsbury v Brighton
Cheltenham v Brentford
Northampton v QPR
Millwall v Bristol Rovers
Watford v Notts County
Rushden & Diamonds v Coventry
Bristol City v Barnet
Colchester v Cardiff
Torquay v Bournemouth
Ipswich v Yeovil
Southend v Southampton
Crystal Palace v Walsall
Swindon v Wycombe
Leyton Orient v Luton
Chesterfield v Huddersfield
Blackpool v Hull
Mansfield v Stoke
Rotherham v Port Vale
Nottingham Forest v Macclesfield
Sheffield United v Boston
Derby v Grimsby
Leeds v Oldham
Wrexham v Doncaster
Scunthorpe v Tranmere
Stockport v Sheffield Wednesday
Bury v Leicester
Rochdale v Bradford
Hartlepool v Darlington
Wolves v Chester
Lincoln v Crewe
Burnley v Carlisle
Reading v Swansea
Gillingham v Oxford
MK Dons v Norwich
Plymouth v Peterborough
Shewsbury v Brighton
Cheltenham v Brentford
Northampton v QPR
Millwall v Bristol Rovers
Watford v Notts County
Rushden & Diamonds v Coventry
Bristol City v Barnet
Colchester v Cardiff
Torquay v Bournemouth
Ipswich v Yeovil
Southend v Southampton
Crystal Palace v Walsall
Swindon v Wycombe
Leyton Orient v Luton