The row started when comments made by an AfD member of the state parliament in Baden-Württemberg about the Holocaust came to light.
In a book published in 2012, Wolfgang Gedeon described the Holocaust as “certain infamous actions” and Holocaust deniers as dissidents.
After he won election to the state parliament in March, the comments resurfaced in the press, with newspapers describing them as a trivialization of the Holocaust and an implicit show of support for Holocaust deniers.
Jörg Meuthen, who along with Frauke Petry is party head of the AfD on the national level, agreed and called on the party chapter in Baden-Württemberg, which he leads, to throw Gedeon out.
If they did not chuck Gedeon out, he would leave the party himself, Meuthen warned.
But the ultimatum did not go down well with co-leader Petry, who accused Meuthen of turning “a professional matter into a personal one”.
The party chapter in Baden-Württemberg were also angered, claiming Meuthen had decided upon Gedeon’s guilt before they had had a chance to properly look into the incident.
With the fledgling party threatened with loosing its second leader within a year - after founder Bernd Lucke was deposed last summer - on Tuesday a compromise was agreed.
After a meeting in Baden-Württemberg, Gedeon agreed to have his party membership suspended until an internal investigation came to a decision on his case in September.
While his demand for Gedeon to leave the party was not met, Meuthen interpreted the decision as a victory.
“I think I have clearly asserted myself,” he said. “I won’t work together with Gedeon any further in this party.”
But according to Die Zeit the dispute is far from settled, with the agreement of 16 of the 23 members of the AfD in the state parliament needed to remove Gedeon from the party.
The row could be more costly for Petry than Meuthen though, writes Die Welt.
Almost the entire party leadership are behind Meuthen, a fact that has as much to do with concerns over Petry's leadership style as over the specifics of the current row.
“It’s touch and go now for Petry,” the conservative daily writes, noting that only one member of the party’s executive committee still supports her.
In state elections in March, the AfD won 15 percent of the vote in Baden-Württemberg, becoming the third largest party in the state parliament.
It also scored double-digit results in Rhineland-Palatinate and Saxony-Anhalt, leading the media to take it seriously as a national party for the first time.
The AfD has been in a series of rows over recent weeks for comments on race and homosexuality by elected officials.
Most recently the party's deputy leader Alexander Gauland suggested that most Germans wouldn't want national football player Jerome Boateng, whose father is from Ghana, as a neighbour.
The remark met widespread disapproval and has been used to explain a recent drop in the AfD's popularity in polling.