Here’s my translation

Alan

Eina nótt var þat, er menn konungssonar sváfu á skipi, en er þeir vakna, sjá þeir, at Björn er horfinn, ok leituðu ok sjá um síðir, at hann hangir upp við siglurá. 
That was one night, that persons (men) of [the] king´s-son slept on [the] ship, but when they awake, they see, that Björn is (has) disappeared, and [they] searched and see at last, that he hangs up against [the] sail-yard.

Þeir vissu ekki, hverju þetta gegndi um líflát Bjarnar, en Gríðr hafði hengt Björn á þeiri nótt fyrir þat hann kallaði Hildi tröllkonu.
They knew not, what this in regard to [the] loss-of-life of Björn meant, but Gríðr had hanged Björn during that (dat fem dem pron, not gen. pers pron.) night for that (ie because) he called Hildr a troll-woman.

Eptir þat siglir Sigurðr frá Finnmark, ok gaf honum þá vel byr ok kom heim til Danmerkr ok hafði fengt of fjár ok gaf gull á tvær hendr. 
After that Sigurðr sails from Finnmörk and [it] well gave him then a fair-wind (ie he got a very fair wind) and came home to Denmark and had obtained an immensity of wealth and gave gold on two hands (ie both left and right, very liberally, cf <höggva á tvær hendr>, under <á>, Z.B.1).

Er hann nú heima með föður sínum. 
He is (stays) now at-home with his father.

Illugi var löngum með Sigurði, en þó átti hann stóran búgarð næri konungshöllinni.
Illugi was a long-time with Sigurðr, but nevertheless he had a large farm near the-king’s-hall.

Litlu síðar tók Hringr konungr sótt þá, er hann leiddi til bana. 
A little later King Hringr took that sickness, which lead him to death.

Sigurðr lætr þegar erfi drekka eptir föður sinn ok lætr til bjóða öllum beztu mönnum í landinu, ok var Sigurðr þá til konungs tekinn yfir þat ríki, er faðir hans hafði átt. 
Sigurðr causes at once to hold (<drekka>, Z2) a funeral-feast for his father, and causes to invite all [the] best people in the-land, and Sigurðr was then elected (<taka til e-s>, Z12) king over that kingdom, which his father had had.

Hringr konungr hafði verit fylkiskonungr í Danmörku, ok hafði hann ráðit fyrir Skán.
King Hringr had been a district-king in Denmark, and he had ruled over Skán (Skåne, Scania).

Signý kemr nú til Danmerkr, ok tók Illugi vel við henni ok þau Hildr. 
Signý comes now to Denmark, and Illugi received her well, and they, Hildr [and he both] (nominative).

Illugi segir þá öll deili á henni. 
Illugi says then (adv) all (n. pl. agrees with <deili>) distinctive-features (ie “everything of note”) about her.

Sigurðr konungr bað hennar sér til handa. 
King Siguðr asked for her (in marriage) on his own behalf (<biðja konu til handa e-m>, under <hönd>, Z1).

Signý segir, at Illugi skuli vera hennar giftingarmaðr. 
Signý says, that Illugi shall be [the] person [man]-to-(agree to)-give-her-away-in-marriage.

Sigurðr talar þá þetta mál við hann, ok með samþykki hennar giftir Illugi honum Signýju. 
Sigurðr speaks (discusses) then this matter with him, and with [the] consent of her, Illugi gives (in marriage) Signý to him (Sigurðr).

Váru þeira samfarir góðar, ok áttu þau mörg börn, ok urðu öll mikils háttar menn.
Their wedded-life-together (marriage, lit: travellings-together, see CV) was good, and they had many children, and all became men of great importance.

Sigurðr konungr ok Signý drottning lifðu mjök lengi, en Illugi lifði lengr, en ekki hefir verit getit barna þeira Hildar. 
King Sigurðr and Queen Signý lived a very long-time, but Illugi lived longer, but [it] has nor been mentioned (<geta>, Z.ii.2) of [the] children of them, Hildr [and Illugi].

Þessi Illugi varð síðan fóstbróðir Gnoðar-Ásmundar.
This Illugi became after-that foster-brother of Gnoð-Ásmundr.

Ok lúkum vér þessi sögu.
And we conclude this story.