Histon's stunning victory over Leeds on Sunday afternoon 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 Liverpool will travel to Preston, Manchester United face a trip to Southampton, Chelsea host Southend and Arsenal entertain Plymouth.
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