The claim in the Constituent Assembly that the Indian Constitution has a 'facile' (easy) amendment procedure refers to the flexibility built into the amending process, particularly when compared to more rigid constitutions like that of the United States. The Indian Constitution provides for three types of amendment procedures:

1. Simple Majority: Certain provisions can be amended by a simple majority of Parliament, similar to ordinary legislation (e.g., creation of new states, changes in citizenship). These are not considered amendments under Article 368.
2. Special Majority (Article 368): Most provisions require a special majority, i.e., a majority of the total membership of each House and a majority of not less than two-thirds of the members of that House present and voting.
3. Special Majority + State Ratification (Article 368 Proviso): For provisions affecting the federal structure (e.g., election of the President, distribution of legislative powers), the special majority in Parliament must also be ratified by the legislatures of not less than one-half of the states by a simple majority.

While the third category is quite rigid, the second category, applicable to a large number of provisions, is considered relatively flexible, leading to the 'facile' claim. This flexibility was intended to allow the Constitution to adapt to changing times and societal needs without resorting to extra-constitutional means.

The Doctrine of Basic Structure and its Limitation on Amending Power:

Yes, the Doctrine of Basic Structure significantly limits the amending power under Article 368. This doctrine, propounded by the Supreme Court in the landmark case of Kesavananda Bharati v. State of Kerala (1973), holds that Parliament's power to amend the Constitution under Article 368 is not absolute and cannot be used to alter or destroy the 'basic structure' or 'essential features' of the Constitution.

Elucidation of its Limiting Effect:

1. From Unlimited to Limited Power: Prior to Kesavananda Bharati, there was a contentious debate regarding the extent of Parliament's amending power. In cases like Shankari Prasad v. Union of India (1951) and Sajjan Singh v. State of Rajasthan (1965), the Supreme Court had held that Parliament could amend any part of the Constitution, including Fundamental Rights. This was challenged in Golaknath v. State of Punjab (1967), where the Court reversed its earlier stand, stating that Fundamental Rights could not be amended. The 24th Amendment Act (1971) was passed to nullify Golaknath, asserting Parliament's unlimited power. However, Kesavananda Bharati struck a balance, affirming Parliament's power to amend but imposing an inherent limitation.

2. Preservation of Constitutional Identity: The doctrine ensures that while the Constitution can evolve, its fundamental identity and core values remain intact. It prevents Parliament from using its amending power to subvert the democratic, secular, federal, and republican character of the Indian state, or to abolish judicial review, which are considered part of the basic structure.

3. Judicial Review of Amendments: The doctrine empowers the judiciary to review constitutional amendments. If an amendment is found to violate the basic structure, the Supreme Court can declare it unconstitutional and void. This acts as a crucial check on legislative supremacy and reinforces constitutional supremacy.

4. Safeguarding Fundamental Rights and Democratic Principles: Although the 'basic structure' is not exhaustively defined, various elements have been identified by the judiciary over time, including the supremacy of the Constitution, the republican and democratic form of government, the secular character of the Constitution, separation of powers, federal character, judicial review, and the independence of the judiciary. By protecting these, the doctrine safeguards the foundational principles of Indian democracy and the rights of its citizens.

5. Balancing Flexibility and Rigidity: The doctrine effectively transforms the 'facile' amending power into a more constrained one, ensuring that flexibility does not lead to constitutional destruction. It strikes a delicate balance between the need for constitutional adaptability and the preservation of its core values.

In conclusion, while the Indian Constitution's amendment procedure might appear 'facile' in certain aspects, the Doctrine of Basic Structure has profoundly and significantly limited the amending power under Article 368. It has established that Parliament, though possessing vast amending powers, cannot use them to destroy the very essence or identity of the Constitution, thereby upholding constitutionalism and the rule of law in India.