Rewire existing subpanel

Hello all. I have an electrical code question.
I have an existing sub panel in a house about 15 feet from my main and meter. The existing sub panel is an old fuse type box with main lugs.
I plan to replace the sub panel with a new panel with breakers and main lugs as well. The existing feed line is in good condition and is 6-6-6 with no ground.
I am replacing the panel with a 125 amp panel and the breaker feeding this sub panel is 50 amps. (I read somewhere that #6 can support up to 55 amp or 60 amp protected?)
My question is: Can I run a #6 ground thhn to the box without replacing the existing feeds and still be code compliant? Or does the ground need to be armored? I have searched and searched but I cannot find any info on this.
TIA -A
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Guy Noir wrote:

trench. You cannot run a separate conductor as you can for a replacement receptacle. -- Tom Horne
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green) or it can be bare #6. Your conductors also need to be in conduit from the main box to the sub. Make sure the neutral is not bonded to ground in the sub.
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the current feed is just pvc sheathed and not in conduit (Think romax but I don't know the proper terminology.)
Can I run the #6 THHN wire separately since it is not a load or neutral but just a ground? Yes, I will not bond the neutral to the ground in the subpanel.
Thanks for the input and excellent suggestions.
-A
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Guy Noir wrote:

Sorry but no. Romex is a brand of cable type NM for Non Metallic. A circuit wired with NM must have all of the conductors inside the jacket of the cable. -- Tom Horne
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So I have taken all of your advice and I will be pulling a new 6/3 with ground. The stuff at the home improvement store is 6-6-6 with a #10 ground. I only want to do this once. Will a #10 ground suffice for national code or should I look for something else?
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Guy Noir wrote:

of over current device. Look it up in the index under Equipment Grounding Conductor, size. There will be a table there that has the answer you need. -- Tom Horne
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