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Keeper Lee Butcher produced a string of excellent second-half saves as Welling reached the second round of the FA Cup for the first time since 1989.
Jake Gallagher's first goal of the season maintained Welling's impressive home form on Saturday.
Lee Clarke's second-half header sent Welling into the first round of the FA Cup on Saturday.
Gillingham have appointed Peter Taylor as their new manager, on an interim basis.
Welling bagged three points at home to Tamworth on Saturday to move into the top six.
Welling put an end to their poor record at Braintree after coming from behind to win on Tuesday night.
Welling were held to a draw by 10-man Barnet in the Skrill Premier on Saturday.
Sittingbourne joint-boss Matt Wyatt believes his team are relishing their role as underdogs in the FA Cup.
Welling moved up to seventh in the Skrill Premier after beating Aldershot in the first-ever meeting between the two clubs on Tuesday night.
Joe Healy scored the only goal as Welling won at Hyde in the Skrill Premier on Saturday.
Welling recover from letting a two-goal lead slip to thrash Forest Green 5-2 in the Skrill Premier.
Welling suffered their first home league defeat in 17 months at the hands of Kidderminster on Saturday.
Two second-half goals saw Welling slip to defeat at Hereford in the Skrill Premier on Saturday.
Joe Healy missed an injury-time penalty as Welling were held to a goalless draw by Salisbury on Monday.
Welling raced into a three-goal lead within the first half-hour as they won at Woking on Saturday.
Welling recorded their first win in The Skrill Premier on Saturday thanks to a late winner at Park View Road
Welling got off the mark in The Skrill Premier on Wednesday with a point against much-fancied Cambridge at Park View Road.
Colin Boswell steps down as the chief executive of the Kent Youth League on Thursday, more than three decades after he founded the league.
Welling boss Jamie Day has started work on bringing in a new defender after Anthony Acheampong left to join Barnet.
Darren Stevens reached three figures to help Kent secure an LV= County Championship draw with Glamorgan at Canterbury on Saturday.
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