Is there a difference in meaning between these two sentences? (Order of modifiers and number+counting word)

I started learning Mandarin not too long ago, starting with the Pimsleur audio course, among other things. One of the sentences you learn is:

"I work for a company in Beijing."

我在北京的一家公司工作。(Wǒ zài běijīng de yījiā gōngsī gōngzuò.)

For some reason, I'm naturally inclined to repeat this as:

我在一家北京的公司工作。(Wǒ zài yījiā běijīng de gōngsī gōngzuò.)

I keep forgetting that 北京的 (běijīng de) comes before 一家 (yījiā).

What is the difference in meaning between the two? Is mine (the second one) wrong or ungrammatical?

谢谢

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