MOSCOW, July 29 (Reuters) - The Russian rouble weakened on Monday, while shares in Yandex, known as Russia's Google, fell on concerns that Moscow is considering limiting foreign investors' share in Russia's "significant" internet companies.

The rouble was down 0.3% against the dollar at 63.54 , hovering at its weakest level since July 10 as of 0750 GMT.

Versus the euro, the rouble was 0.2% lower at 70.67 , heading away from its strongest level since March 2018 of 69.95 that it briefly touched last week.

The central bank's decision to cut its key rate to 7.25% on Friday caused no visible market reaction.

But market sentiment soured somewhat after more than 1,000 people were detained in Moscow on Saturday in one of the biggest crackdowns of recent years against an increasingly defiant opposition decrying President Vladimir Putin's grip on power.

Investors ignore political risks linked to rising discontent for now but such risks could become "the black swan" should they increase sharply, said Kirill Tremasov, head of research at Loko-Invest and a former research department chief at the economy ministry.

Yandex, Russia's leading internet firm, was in the spotlight after its shares rallied on Friday on strong corporate earnings but soon fell sharply on reports that the lower house of parliament was mulling limiting foreign ownership in "significant" IT firms to 20%.

Yandex, whose shares are listed in New York and Moscow, said the proposed bill could "demolish a unique ecosystem of internet business in Russia where local players successfully compete with global companies".

"Friday night's headlines of proposals to limit foreign ownership in online companies, particularly Yandex, are likely to weigh on sentiment in the sector," Alfa Bank said in a note.

Yandex shares were down 2.4% on the Moscow Exchange, underperforming the benchmark MOEX stock index that was 0.2% higher at 2,720.2.

The dollar-denominated RTS index was flat at 1,148.4 points.

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(Reporting by Andrey Ostroukh; Editing by Dale Hudson)