http://www.thefreedictionary.com/despise

This ref has says that both Jim and Robin are right

Is 中辞典 not just an abridged version of GG5? Regardless, I disagree with them.
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Oxford: "Feel contempt or a deep repugnance for"
Cambridge: "to feel a strong dislike for someone or something because you think that that person or thing is bad or has no value"
Macmillan: "to hate someone or something and have no respect for them"
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Here are the 3 top example sentences from Oxford, Cambridge, and Merriam-Webster's entries for despise.
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"He despised himself for being selfish"
"She despised him for the way he treated her sister"
"She was despised as a hypocrite"
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Compare with the examples sentences for 侮る in daijs, prog, and E-DIC
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簡単な問題だと侮ってはいけない
侮って油断するな
子供の好奇心を侮ってはいけない
侮れない相手だ
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侮る would not be used to translate any of the English sentences, nor "despise" for the Japanese ones.

GG5 & 中辞典 - both have "despise" as the first gloss.

I'm not sure despise => hatred. For example Wordnet says of it:
v : look down on with disdain; "He despises the people he has to work for"; "The professor scorns the students who don't catch on immediately" [syn: {contemn}, {scorn}, {disdain}]

daijr/s

Reordered glosses. To me, "despise" implies hatred, which isn't part of the definition.

@@ -18 +17,0 @@
-<gloss>to despise</gloss>
@@ -19,0 +19 @@
+<gloss>to look down on</gloss>
@@ -22 +21,0 @@
-<gloss>to look down on</gloss>
@@ -23,0 +23 @@
+<gloss>to despise</gloss>