Preston North End will face Liverpool in the third round of the FA Cup.
Alan Irvine's team were drawn against the joint Premier League leaders in Sunday's draw.
Striker Neil Mellor will get the chance to face his former club - he made 12 appearances and scored two goals for the Anfield giants between 2002 and 2006.
Histon's stunning victory over Leeds did not bring them the FA Cup third-round rewards they were hoping for.
The Blue Square Premier promotion chasers were 1-0 winners over Coca-Cola League One giants United in the second round, but were visibly disappointed to be handed a home tie with Swansea as they watched the live draw.
Fellow non-leaguers Blyth Spartans will host Barclays Premier League side Blackburn if they win their replay against Bournemouth.
FA Cup holders Portsmouth were drawn at home to Bristol City while Arsenal will entertain Plymouth.
Manchester United go to Southampton, Manchester City are at home to Nottingham Forest, Tottenham Hotspur take on Wigan Athletic and Chelsea face Southend.
If Morecambe overcome Cheltenham when the tie is finally played after Saturday's first attempted was abandoned, they will be at home again - to Championship outfit Doncaster.
Full draw:Portsmouth v Bristol City
Sheffield Wednesday v Fulham
Preston v Liverpool
Birmingham v Wolves
West Ham v Barnsley
Middlesbrough v Barrow
Hull v Newcastle
Hartlepool v Stoke
Chelsea v Southend
Manchester City v Nottingham Forest
Cardiff v Reading
Ipswich v Chesterfield or Droylsden
Charlton v Norwich
West Brom v Peterborough or Tranmere
Torquay v Blackpool
Leyton Orient v Sheffield United
Southampton v Manchester United
Millwall v Carlisle or Crewe
Histon v Swansea
Forest Green v Derby
QPR v Burnley
Leicester v Crystal Palace
Tottenham v Wigan
Morecambe or Cheltenham v Doncaster
Arsenal v Plymouth
Notts County or Kettering v Eastwood
Bournemouth or Blyth Spartans v Blackburn
Macclesfield v Everton
Watford v Scunthorpe
Sunderland v Bolton
Coventry v Kidderminster
Gillingham or Stockport v Aston Villa
:: Ties to be played on the weekend of January 3 and 4.
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